Review of Elementary Algebra Content

Size: px
Start display at page:

Download "Review of Elementary Algebra Content"

Transcription

1 Review of Elementar Algebra Content 0

2 1

3 Table of Contents Fractions...1 Integers...5 Order of Operations...9 Eponents...11 Polnomials...18 Factoring... Solving Linear Equations...1 Solving Linear Inequalities... Graphing Linear Equations...8 Finding the Equation of a Line... Application of Linear Equations...8 Functions...50 Sstems of Linear Equations...5 Application of a Sstem of Linear Equations...6 Solving Linear Inequalities in Two Variables...6 Solving Sstems of Linear Inequalities...67 Solving Absolute Value Equations...7 Solving Absolute Value Inequalities...76 Vocabular Used in Application Problems...79 Solving Application Problems...81 Comprehensive Review of Elementar Algebra...88 i Revised Ma 01

4

5 FRACTIONS Lowest Terms Eample: 1 Simplif: 1 = 15 5 = 5 Addition/Subtraction 1 Rule: To add and subtract fractions ou need a Common Denominator. Eample: + + = = 7 Eample: = + 8 = = 1 For this eample, we alread have the common denominator. This eample, we need to find a common denominator, or a denominator which is DIVISIBLE b and. One such denominator is 1. Eample: = = 1 1 = 1 1 = Note: An number over itself is equal to 1. When ou multipl a number b 1, ou are not changing the number, just renaming it. 9 = 1 = 1 = 1 9 1

6 FRACTIONS Multiplication/Division Rule: Multiplication and division of fractions do not require a common denominator. Eample: = = 5 1 = 10 Cancel a common factor of Multipl across Eample: 5 6 = = 5 = 15 = 5 To divide b 5, multipl b the 6 reciprocal which is 6 5

7 FRACTIONS Problems Answers Perform the following operations

8 FRACTIONS Problems (continued) Answers Perform the following operations

9 INTEGERS Definition: The absolute value of a number is the distance between 0 and the number on the number line. The smbol for The absolute value of a is a. The absolute value of a number can never be negative. Eamples: 1) = ) = Addition of Signed Numbers: Like signs: Add their absolute values and use the common sign. Eamples: 1) + ( ) = ( + ) = 5 ) + 8= ( + 8) = 1 Unlike signs: Subtract their absolute values and use the sign of the number with the larger absolute value. Eamples: 1) + =+ ( ) = 1 ) + ( 6) = ( 6 ) = 5

10 INTEGERS Subtraction of Signed Numbers 1) Change the subtraction smbol to addition. ) Change the sign of the number being subtracted. ) Add the numbers using like signs or unlike signs rules for addition. Eamples: 1) = + ( ) = 7 ) 6 ( ) = 6+ ( ) = 8 ) ( ) = + = Multiplication and Division of Two Signed Numbers: Like signs: The product or quotient of two numbers with like signs is positive. Eamples: 1) ( )( ) = 1 ) 8 = ) 16 = = Unlike signs: The product or quotient of two numbers with unlike signs is negative. Eamples: 1) ( )( ) = 1 ) 15 = 5 Division b 0 is undefined. Eample: 0 is undefined. 6

11 INTEGERS Addition and Subtraction of Signed Numbers Problems Answers Perform the following operations ( ). 6+ ( ) ( 1) ( 5) 1 7. ( 11) ( ) ( ) ( 8) ( 1) ( 9 ) ( ) + 7+ ( 6) ( 1 + 1) ( 1 9)

12 INTEGERS Multiplication and Division of Signed Numbers Problems Answers Perform the following operations. 1. ( )( ) ( 10)( 1) ( 9.8) ( 7) ( 5.1 )(.0) ( ) 10. ( 9 1)( ) ( 6) 6 8

13 ORDER OF OPERATIONS Please Ecuse M Dear Aunt Sall Grouping Smbols Parentheses ( ) Brackets [ ] Square Roots Absolute Value Eponents Multipl Divide "as ou see it, left to right" Add Subtract "as ou see it, left to right" Eamples: Note: When one pair of grouping smbols is inside another pair, perform the operations within the innermost pair of grouping smbols first. 6 ( 5 ) ( ) ( ) ( ) [ ] Eamples: Note: For a problem with a fraction bar, perform the operations in the numerator and denominator separatel = = = = ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) = = = = =

14 ORDER OF OPERATIONS Problems Answers Perform the following operations ( + ) ( ) ( ) 8. ( ) ( )

15 EXPONENTS Laws of Eponents Definition: n a = a a a... a n times a = the base n = the eponent Properties: 1. Product rule: m n m n a a = a + Eample: + a a = a a a a a = a = a 5 times times. Quotient rule: a a m n m n = a Eample: a a aaaaaaa = = a = a aaaaa 7 5 n. Power rule: ( ) m a nm = Eample: ( a ) = a = a 6 a Raising a product to a power: ( ) n n n ab = a b Raising a quotient to a power: n a a = b b n n. Zero power rule: 0 a = 1 Eample: ( ) 0 ( ) = 1; = = 1 1 = 1 Eamples: ( )( ) ab a b c = a b c = a b c 8 16 = = 81 1 ( ) ( )( ) ( )( ) + 5 = + 5 = =

16 EXPONENTS Problems Answers Perform the following operations and simplif ( )( ) ( 5ab 6 )( 7ab 8 ) 9 9 5a b z z. 7 10ab ab 5ab 5. ( ab c ) 1 6 ab c 6. 5 ( ) z z 7. ( ) mn 9 mn 8. 0 ( n ) 9 9. ( 5 ) ( ) z 8 z 6 1

17 EXPONENTS Negative Eponents Definitions: a m m a 1 = 1 a m 1 a = a m a 1 a b m n m b a m n m Note: The act of moving a factor from the numerator to the denominator (or from the denominator to the numerator) changes a negative eponent to a positive eponent. Eamples: = = , 000 ab 1 1 ab a b a a = = = 1 1 b b Since b has a negative eponent, move it to the denominator to become b 5 5 z = = z 5 z Onl the factors with negative eponents are moved. 1

18 EXPONENTS Eample: 1 1 = Within the parenthesis, move the factors with negative eponents. = 6 Simplif within the parenthesis. ( ) = = 6 ( ) ( ) 1 Use the power rule for eponents. 1 = Move all factors with negative eponents. ( ) 16 1 = Simplif. 1

19 EXPONENTS Problems Answers Perform the following operations and simpl. 1. ( ) = = ( a 0 b )( ab c ) 7 9ab c 6. ( ebc ) ( ab 1 ) 18ac e 11 b 7. ( ) ab c ( a) 0 1 bc 18a c b 15

20 EXPONENTS Problems (continued) Answers a a a 1. 6 ( a aa ) b 5 b 15. a 6 a 1 a 1 7 a b a 16. p 5 p 6 p ab ab ab

21 EXPONENTS Problems (continued) Answers 8. b c bc 1 bc a 6 a 9 a 6 5. ( ) 5 6. ( ) a b 0 a 15 b 7. 5 ( ) 1 a a b ( a b ) ( a b) a b ab 9. 5 ( ) a b 0. 5 ( ) a b a b ( ) 6 5a ab 5. ( ab ) ( a b ) 5. a ( a ) ( a ) b 16a a b b 19 a a b 7a 16 17

22 POLYNOMIALS n Definition: A sum of a finite number of terms of the form: a n where a n is a real number and n is a non-negative integer. (No negative eponents, no fractional eponents.) Eamples: is a polnomial + is not a polnomial Tpes of Polnomials: Monomial: A polnomial with 1 term Eample: Binomials: A polnomial with terms Eample: + Trinomial: A polnomial with terms Eample: + Degree: Degree of a term the sum of the eponents on the variables. Eample: has degree + = 7 Degree of a polnomial the largest degree of an of the terms. In a polnomial with one variable it is the largest eponent. Eample: has degree + has degree since has degree + 1= 18

23 POLYNOMIALS Addition of Polnomials Adding Polnomials: Combine like terms. Eample: ( ) ( ) = = 5 Subtraction of Polnomials Subtracting Polnomials: Distribute the negative sign and combine like terms. Eample: ( ) ( ) = = Multiplication of Polnomials Multipling Polnomials: Use the distributive propert. Eample: ( + )( + ) = ( + ) + ( + ) = = + + Eample: ( + )( ) = ( )( ) + ( )( ) + ( ) + ( ) = = Eample: ( )( ) + The multiplication of the sum and difference of two terms. ( ) ( ) ( ) ( ) = = = + = 9 16 The answer is the difference of two squares. 19

24 POLYNOMIALS Eample: ( + ) The square of a binomial. ( + )( + ) = ( ) + ( ) + ( ) + ( ) Eample: = = ( ) 16 9 = + + The answer is a perfect square trinomial. ( a b) The cube of a binomial. ( a b)( a b)( a b) = ( a b)( a b) ( a b) = aa + a( b) b( a) b( b) ( a b) ( ) = + a ab ab b a b ( ) ( ) ( ) ( ) = a ab + b a b = + a a b ab a b b a b = a a b a b + ab + ab b = a a b + ab b Use the distributive propert. Division of Polnomials Dividing a polnomial b a monomial. Eample: = + 1 = = 8 + =

25 POLYNOMIALS Problems Answers Find the degree of the following and determine what tpe of polnomial is given Trinomial, degree 5. 5 Binomial, degree 5. Monomial, degree 5. 8 Monomial, degree 0 Add: 5. ( + 5+ ) + ( 6 8) Subtract: 6. ( + 5+ ) ( 6 8) Multipl: 7. ( + )( 5+ ) 8. ( + )( ) 9. ( + )( ) 10. ( 5 + 7) 11. ( 9 ) 1. ( ) Divide:

26 FACTORING Factoring Out the Greatest Common Factor 1. Identif the TERMS of the polnomial.. Factor each term to its prime factors.. Look for common factors in all terms.. Factor out the common factor. 5. Check b multipling. Eample: Factor Terms: 6, 8, and + Factor into primes + Look for common factors Factor out the common factor Answer: ( + ) Check: ( ) ( ) ( ) + Check b multipling Factoring Trinomials of the Form: + b + c To factor , look for two numbers whose product = 6, and whose sum = 5 List factors of 6 Choose the pair which adds to 5 and multiplies to = = 6 Since the numbers and work for both, we will use them. So, = ( ) ( ) ( )( ) = = + + Substitute + in for 5. Factor b grouping. Answer: ( + )( + ) Check b multipling: ( )( ) + + = =

27 FACTORING Factoring Trinomials of the Form: a + b + c Eample: Factor Factor out the GCF ( ) GCF is. Identif the values of a, b, and c for the epression + 7+ a= ; b= 7; and c =. Find the product of ac ac= = 6. List all possible factor pairs that equal ac and identif the pair whose sum is b. 5. Substitute boed values from Step in for the b term =7 + = & 6 and & are the onl factor pairs. 1 & 6 is the pair that adds up to 7 and multiplies up to 6. Substitute for Group the two pairs First step in factor b grouping. 7. Factor the GCF out of each pair. ( 1) ( 1) 8. Factor out ( + 1) ( 1)( ) The resulting common + 1 factor is ( ) + + These are the two factors of Recall the original GCF from Step 1. Answer: ( + + ) = ( + )( + ) 7 1 ( + )( + ) 1 There are factors of Check b multipling. ( )( ) =

28 FACTORING Eample: Factor Note: If a cis negative, the factor pairs have opposite signs. 1. Factor out the GCF. There is no GCF.. Identif the values of a, b, and c. a= ; b= ; c=. Find the product of ac. ( ) ac= = 1. List all possible factor pairs that equal ac and identif the pair whose sum is b. ( ) ( ) = 11 Include all possible factor combinations, including 1 ( + 1) 1+ ( + 1) =+ 11 the + / combinations since ac is negative. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = Substitute the boed values from Step in for the b term. 6 = + Substitute 6 in for 6. Group the two pairs. = + 6 First step in factor b grouping. 7. Factor the GCF out of each pair ( ) ( ) = + + The resulting common +. factor is ( ) 8. Factor out ( + ) Answer: ( + )( ) ( + )( ) These are the factors. 9. Check: ( )( ) + =

29 FACTORING Factoring Perfect Square Trinomials Eample: Factor STEP 1. Determine if the trinomial is a Perfect Square. Is the first term a perfect square? NOTES Yes = ( ) Is the third term a perfect square? +81 Yes ( ) Is the second term twice the product of the square roots of the first and third terms? 9 9 = 9 = Yes ( 9) = 18. To factor a Perfect Square Trinomial: Find the square root of the first term = Identif the sign of the second term + Find the square root of the third term 9 81 = 9 Write the factored form = = + 9 Answer: ( 9) + Check our work ( )( ) ( )( ) ( ) = Eample: Factor STEP 1. Is the trinomial a Perfect Square Trinomial? Is the first term a perfect square? Is the third term a perfect square? Is the second term twice the product of the square roots of the first and third terms?. Factor the trinomial Square root of the first term 5 = 5 Sign of the second term Square root of the third term = Factored form Answer: NOTES 5 Yes ( ) 5 = 5 Yes ( ) = 0 Yes ( ) 5 = = ( 5+ )( 5+ ) = ( 5+ ) ( 5+ ) Check our work ( )( ) =

30 FACTORING Factoring the Difference of Two Squares Eample: Factor 6 STEP NOTES 1. Are there an common factors? No. Determine if the binomial is the Difference of Two Squares Is the first term a perfect square? Yes = Is the second term a perfect square? 6 Yes 6 6 = 6 Is this the difference of the two terms? 6 Yes, means the difference of the two terms.. To factor the Difference of Two Squares: Find the square root of the first term. Find the square root of the second term. 6 6 = 6 = Write the factored form. Answer: 6 = ( + 6)( 6) ( + 6)( 6) Check our work. ( )( ) = 6 The factors are the product of the sum (+) and difference ( ) of the terms square roots. Eample: Factor 6a 11b STEP NOTES 1. Are there an common factors? No. Is the binomial a Difference of Two Squares? Is the first term a perfect square? 6a Yes 6a 6a= 6a Is the second term a perfect square? 11b Yes 11b 11b= 11b Is this the difference of the two terms? 6a 11b Yes, means the difference of the two terms.. To factor the Difference of Two Squares: Find the square root of the first term. 6a 6a = 6a Find the square root of the second term. 11b 11b = 11b Write the factored form. ( )( ) 6 11 = a b a b a b Answer: ( 6a+ 11b)( 6a 11b ) Check our work. ( )( ) 6a+ 11b 6a 11b = 6a 11b The factors are the product of the sum (+) and difference ( ) of the terms square roots. 6

31 FACTORING FACTORING STRATEGY 1. Is there a common factor? If so, factor out the GCF, or the opposite of the GCF so that the leading coefficient is positive.. How man terms does the polnomial have? If it has two terms, look for the following problem tpe: a. The difference of two squares If it has three terms, look for the following problem tpes: a. A perfect-square trinomial b. If the trinomial is not a perfect square, use the grouping method. If it has four or more terms, tr to factor b grouping.. Can an factors be factored further? If so, factor them completel.. Does the factorization check? Check b multipling. 7

32 FACTORING STRATEGY Alwas check for Greatest Common Factor First Then continue: Terms Terms Terms Difference of Two Squares ( ) ( ) + ( )( ) Perfect Square Trinomials ( ) + ( )( ) + ( + 5)( + 5) ( + 5) 5 5 Form: + b + c 6+ 5 List factors of Choose the pair which adds to = 6 ( 5)( 1) 8 Form: a + b+ c = 0 List factors of Choose the pair which adds to = 7 Substitute this pair in for the middle term Factor b Grouping. ( 5 + ) ( 5 + ) ( 5+ )( ) Factor b Grouping ( + ) ( + ) 7 6 ( + )( 7 6)

33 FACTORING Problems Answers Factor completel ( + )( + ). u + 15u+ 56 ( u+ 8)( u+ 7) ( + 10)( ). 6 0 ( 10)( + ) 5. m 15m+ 5 ( m 9)( m 6) ( + 7)( ) 7. a 16ab 8b + + ( a+ 1b)( a+ b) ( 6)( + ) 9. 1 ( 7+ )( 1) ( 5)( ) 9

34 FACTORING Problems (continued) Answers ( 1)( + ) ( )( + 1) ( + + ) ( + )( 1) ( 1)( + 1) ( + 9)( + )( ) ( + )( ) ( + 5)( + ) ( 9) ( + ) ( + ) 0

35 SOLVING LINEAR EQUATIONS To Solve a Linear Equation 1. Remove grouping smbols b using the distributive propert.. Combine like terms to simplif each side.. Clear fractions b multipling both sides of the equation b the Least Common Denominator.. Move the variable terms to one side and the constants to the other side. Do this b adding or subtracting terms. 5. Solve for the variable b multipling b the inverse or dividing b the coefficient of. 6. Check b substituting the result into the original equation. + 8 = + Eample Solve ( ) ( ) Solution: 1 8= + 0 = + = = = 8 Use the distributive propert. Combine terms Move variable to one side, numbers to the other Divide to solve for the variable. Check: ( ) ( ( ) + ) = ( ) + ( 1) ( ) = + 6 ( 8) = = 8 Eample: Solve 1 = Solution: Check: 1 ( ) ( ) = ( ) = = 8 8 = = 8 = = = Clear fractions b multipling b the LCD

36 SOLVING LINEAR EQUATIONS Special Cases: 1. When the variable terms drop out and the result is a true statement, (i.e., = or 0= 0), there are an infinite number of solutions. The equation is called an identit. Eample: Solve ( ) ( ) + 9 = = = = 9+ 0= 0 Infinite number of solutions Solution: All real numbers. When the variable terms drop out and the result is a false statement (i.e., 10 = or 8 = 0), there is no solution. The equation is called a contradiction. Eample: Solve ( ) 5= + 5= 6+ 5= 5= Solution: No solution

37 SOLVING LINEAR EQUATIONS Problems Answers Solve. 1. ( 5) ( 5) + = 17 =. ( 1 )( 5) ( 1 )( ) = + = = No solution. + = + 6 Infinite number of solutions: All real numbers 5. 5 ( ) + + = + = 0

38 SOLVING LINEAR INEQUALITIES Solving Linear Inequalities Linear inequalities are solved almost eactl like linear equations. There is onl one eception: if it is necessar to divide or multipl b a NEGATIVE number, the inequalit sign must be reversed. The solution can be written as a statement of inequalit or in interval notation. It can also be shown as a graph. In interval notation and graphing: Use a bracket, [ ], or closed circle, if the endpoint is included in the solution Use parenthesis, ( ), or open circle, if the endpoint is not included in the solution Eample: 1 < Interval Notation: (,1) Eample: ( ) or Distribute Move variable term to one side, constants to the other 11 Divide b and reverse the inequalit because of division 11 Simplif Solution: Interval Notation:, [ or 11 5

39 SOLVING LINEAR INEQUALITIES Problems Answers Solve. 1. a ( a ) 7> a < or (, 10). ( ) ( ) or [, ). 5 ( ) + + < + 0 > or ( 0, ) or (,1] 5. ( ) ( ) or [, ) 6. ( ) ( ) 5 + < < or (, 11) 5

40 Compound Inequalities SOLVING LINEAR INEQUALITIES Compound inequalities are solved the same wa as simple inequalities. An operation (addition, subtraction, multiplication, division) must be performed on all three pieces of the inequalit. Never remove the variable from the middle piece of the inequalit. Alwas remember that when dividing or multipling b a NEGATIVE number, the inequalit sign must be reversed. Eample: Solve 8 < < < < Add to all three parts of the inequalit Divide all three parts of the inequalit b Simplif In interval notation, this is (, ], because is between and, including the endpoint, but not including the endpoint. Solution: < Interval Notation: (,] ( or Eample: Solve 5 < < > > 5 > > 5 < < 5, When dividing an inequalit b a negative number, reverse the inequalit sign. Keep the variable in the middle piece of the inequalit. Arrange the inequalit so that the lesser value is on the left and the greater value on the right, and change the inequalit smbols to preserve the relationship. Solution: 5 5 < < Interval Notation:, ( ) -5-0 or

41 SOLVING LINEAR INEQUALITIES Problems Answers Solve 1. < + 1< 1 1 < < or ( 1,1) or [ 5 ],0. < + 8 < or (, ]. < 5 < < < or ( 1,1 ) 5. 1 < < or ( 5, 1] 6. < < 0 < < or ( ), 7. 0< 1 < 7 1 < < or ( 1, ) or 7 [, ] 9. 7< < 11 < < or 11 (, ) 10. < < 6 < < 0 or ( ),0 7

42 GRAPHING LINEAR EQUATIONS Standard Form: a + b = c A line is made up of an infinite number of points in the form (,). The coordinates for each of these points will satisf the equation for the line (make it true). Two special points on the line are its and intercepts. These are the points on the line where the line crosses the ais and the ais. The intercept is in the form (,0), and the intercept is in the form (0,). Find the intercept of the line b substituting 0 for, and solving for, (,0). Find the intercept of the line b substituting 0 for, and solving for, (0,). Eample: Graph the line using the intercept method = 1 Find the intercept: Substitute 0 for and solve for : (,6 ) = 1 ( ) 0 = 1 0= 1 = 1 1 = = The intercept is (,0). Find the intercept: Substitute 0 for and solve for : = 1 ( ) 0 = 1 0 = 1 = 1 1 = = The intercept is (0,). 0 To graph the line, graph these two points, and connect them. As a check, find one more point on the line. Let = and find = 6. The point (,6) is on the line and collinear with (,0 ) and ( 0, ). 8

43 GRAPHING LINEAR EQUATIONS Slope Intercept Form: = m + b If the equation of the line is in slope-intercept form, we use the slope m, and the intercept, (0,b), to graph the line. The slope of a line passing through points (, ) and (, ) vertical change rise change in 1 1 is 1 m= = = = if 1 horizontal change run change in 1 Eample: Graph the line: = +. The slope calculation is called a Rate of Change. The rate of change describes how one quantit (the numerator) changes with respect to another (the denominator). Identif the slope as a fraction: In the equation = m + b, m is the slope. In the equation = +, is the slope, so m =. 1 Identif the intercept as a point: In the equation, m b, ( 0, b) = + is the intercept. In the equation, = +, (0,) is the intercept Graph the intercept: (0,) is the first point to be graphed 8 Graph the slope: rise of m = = 1 run of 1 is positive; move up (rise) from the intercept, (0,), to the point (0,5). 6 (0,) (1,5) is positive; then move 1 to the right (run) from (0,5) to (1,5). (1,5) is the second point of the line. Graph the line through the two points: Graph the line through the points (0,) and (1,5). 9

44 GRAPHING LINEAR EQUATIONS Special Cases: The graph of the linear equation the( 0, k ) Eample: = = k, where k is a real number, is the horizontal line going through The graph of the linear equation = k, where k is a real number, is the vertical line going through the point ( k,0). Eample: = 0

45 GRAPHING LINEAR EQUATIONS Problems Graph the following equations: 1. 5 = 5. + = 5. = +. = 5. 6 = 1 6. = 1

46 GRAPHING LINEAR EQUATIONS Answers 1. 5 = 5. + = 5. = +. = 5. 6 = 1 6. = 5

47 FINDING THE EQUATION OF A LINE The standard form of the equation of a line is written as a + b = c where a and b are not both 0. Definition of slope: Let L be a line passing through the points (, ) and (, ) of L is given b the formula: m = Then the slope The slope-intercept form of the equation of a line is written as = m + b where m is the slope and the point ( 0,b) is the -intercept. Point-Slope Form: The equation of the straight line passing through ( 1, 1) is given b = m( ). 1 1 and having slope m Parallel Propert: Parallel lines have the same slope. Perpendicular Propert: When two lines are perpendicular, their slopes are negative (opposite) reciprocals of one another. The product of their slopes is 1.

48 FINDING THE EQUATION OF A LINE To find the equation of a line, use: Slope-Intercept form: m b m Slope-Intercept Form = +, slope m, intercept ( 0,b ) or Point-Slope form: = ( ), slope m, point (, ) 1 1 When given the slope and intercept: use the slope-intercept form = m + b 1 1 Eample: Find the equation of the line with slope and intercept ( 0, ) Use the slope-intercept form, = m + b, and fill in the values: Solution: The equation of the line is: = Make the substitutions of m = slope is and b= intercept is 0, ( ( )) Point-Slope Form When given the slope and a point on the line: use the point-slope form = m( ) Eample: Find the equation of the line with slope, through the point ( 6,1 ) Use the point-slope form, = m ( ) 1 1, and fill in the values = ( 6) 6 1 = 1 1= 1+ 1= + 1 = Solution: The equation of the line is Make the substitutions of m = slope is and = 6 and = 1 point 6,1 ( ( )) 1 1. =

49 FINDING THE EQUATION OF A LINE When given two points on the line, first find the slope m, then use the slope and either one of the = m points to find the equation using the point-slope form ( ) 1 1 Eample: Find the equation of the line through the points (, ) and ( 1,1 ). 1 Slope m = = = 1 1 ( ) ( ) = m 1 1 = = = 1 Make the substitutions of m= slope is and = and = ( ) ( point (, )). 1 1 Solution: The equation of the line is = 1 5

50 FINDING THE EQUATION OF A LINE Problems Answers 1. Put the following in slope-intercept form. a. 8 = = 1 b. + = = + 1. Find the slope. a. (, ) and ( 7,9 ) 5 b. (,1) and (, ) 5. Find the equation of the line through ( ) and parallel to = , = + 1. Find the equation of the line through ( 1, ) and perpendicular to =. 1 1 = 6

51 FINDING THE EQUATION OF A LINE Problems Answers Find the equation of the line that satisfies the given conditions. Write the equation in both standard form and slope-intercept form. 5. slope = 1 line passes through ( ) Standard Form, = 5 Slope-Intercept Form 1 5 = + 6. slope = 5 line passes through ( ) 6 0, = 0 5 = 6 7. slope = 1 -intercept = = 8. horizontal line through ( 1, ) = 9. slope is undefined and passing through ( 5,6) = slope = -intercept 5 + = = 11. horizontal line through ( 5, ) = 1. vertical line passing through ( 5, ) = 5 1. line passing through ( 1, ) and ( ) 1. line passing through (, ) and ( 5, 1) 5, = 5+ 8 = 17 1 = = line passing through (, ) and (, 7) = 16. line parallel to + = 6and passing through + = 5 = + 5 ( 1, ) 17. line passing through ( 5, ) and parallel to = = line perpendicular to + 5 = and 5 = 9 passing through ( 1, 7 ) 5 9 = + 7

52 APPLICATION OF LINEAR EQUATIONS Eample: Determine Rate of Change A credit union offers a checking account with a service charge for each check written. The relationship between the monthl charge for each check and the number of checks written is graphed below. At what rate does the monthl charge for the checking account change? Also, find the unit cost, another wa to epress the rate of change. 0 Monthl Charge for checking account ($) (50, 1) (75, 16) Number of checks written during the month Find the rate of change (slope of the line). The units will be dollars per number of checks. From the graph, we see that two points on the line are (50, 1) and (75, 16). If we let, = 50,1 and, = 75,16, we have ( ) ( ) ( ) ( ) 1 1 Rate of change ( 1) ( ) 1 ( ) ( ) dollars 16 1 dollars dollars = = = checks checks 5checks The rate of change can be epressed as $ for ever 5 checks. Rate of change can also be epressed as $0.08 per check or 8 per check. The monthl cost of the checking account increases $ for ever 5 checks written. To find the cost of 1 check (unit cost), take the fraction which represents the rate of change, 5, and divide both the numerator and denominator b 5. dollars 5.08 dollars = = = $.08 / check. 5 checks check The unit cost = $.08 per check. 8

53 APPLICATION OF LINEAR EQUATIONS Problems Answers The graph models the number of members in an organization from 001 to 010. Number of Members (001,00).. (007,15) Year 010 How man members did the organization have in 009? 100 members Find the rate of change. (slope of the line) loss of 5 members ever two ears What is the rate of change per ear? loss of 1.5 members per ear 9

54 FUNCTIONS Recall that relation is a set of ordered pairs and that a function is a special tpe of relation. A function is a set of ordered pairs (a relation) in which to each first component, there corresponds eactl one second component. The set of first components is called the domain of the function and the set of second components is called the range of the function. Eamples: Determine if the following are functions. Domain Range This is a function, since for each first component, there is eactl one second component. Domain Range This is not a function since for the first component, 7, there are two different second components, and 5. Vertical Line Tests: A graph in the plane represents a function if no vertical line intersects the graph at more than one point. Eamples: This is a graph of a function. It passes the vertical line test. Since we will often work with sets of ordered pairs of the form (,, ) it is helpful to define a function using the variables and. Given a relation in and, if to each value of in the domain there corresponds eactl one value of in the range, then is said to be a function of. (,) (, ) This is not a graph of a function. For eample, the -value of is assigned two different values, and. (for the first component,, there are two different second components, and.) 50

55 FUNCTIONS Notation: To denote that is a function of, we write = f ( ). The epression f ( ) f of. It does not mean f times. Since and f ( ) are equal, the can be used interchangeabl. This means we can write =, or we can write f ( ) =. is read Evaluate: To evaluate or calculate a function, replace the in the function rule b the given value from the domain and then compute according to the rule. For eample: Eamples: 1. ( ) Given: f = 6+ 5 ( ) ( ) Find: f = = 17 f f ( ) ( ) 0= 60+ 5= 5 ( ) ( ) 1 = = 1. ( ) g = + Given: 5 8 ( ) ( ) ( ) Find: g 1 = = 6 ( ) ( ) ( ) g 0 = = 8 ( ) ( ) ( ) g 1 = = 16 51

56 FUNCTIONS Problems Answers 1. Determine whether or not each relation defines a function. If no, eplain wh not. a. Domain Range a. Yes, for each first component, there corresponds eactl one second component. b. Domain Range b. No, for each first component,, there corresponds two different second components, and c. {( 1, ),(, ),( 1, 0) } d. {(, ),(, ),( 1,) }. Determine the domain and range of each of the following: c. No, for the first component, 1, there corresponds two different second components and 0. d. Yes, for each first component there corresponds eactl one second component. a. {(, ),(, ),( 0,0 ),(,7) } a. Domain: {,,0, } Range: { 0,,,7 } b. b c. c Domain:{ 5, 7, 9} Range: {,,6 } Domain: { 1,5,9 } Range: { 7,1,1 }. Given ( ) ( ), f = + and g = find the following: a. f ( ) a. 0 b. f ( ) b. c. g ( 0) c. d. g ( ) d. 1 5

57 SYSTEMS OF LINEAR EQUATIONS The following summar compares the graphing, elimination (addition), and substitution methods for solving linear sstems of equations. Method Eample Notes Graphing + = 6 You can see the solution is where the two = 8 lines intersect but if the solution does not involve integers it s impossible to tell eactl what the solution is. 5 +=6 The solution for this sstem of linear,. equations is ( ) (,-) -=8 - - Substitution = 1 = Substitute 1 for : ( 1) = Solve for : = Back-substitute: = ( ) 1= 5 Solution: (,5) Gives eact solutions. The solution for this sstem of equations is (,5 ). Elimination (Addition) + = 8 5+ = Multipl the top equation b 5 and the bottom equation b. This will result in opposite coefficients of = = 68 Add the two equations: 7 = 8 = To find, substitute = into either original equation. + ( ) = 8 = 10 10, Solution: ( ) Gives eact solution. Eas to use if a variable is on one side b itself. The solution for this sstem 10,. of equations is ( ) 5

58 Three Possible Solution Tpes SYSTEMS OF LINEAR EQUATIONS 0 No Solution (Parallel Lines) Inconsistent Sstem Independent Equations # = different # (contradiction). 0 Eactl One Solution (point of intersection) Consistent Sstem Independent Equations Solution 0 Infinitel Man Solutions (Lines Coincide-an point on the line is a solution) Consistent Sstem Dependent Equations # = same # (identit) 5

59 SYSTEMS OF LINEAR EQUATIONS Solving Linear Sstems B Substitution Method To solve a linear sstem of two equations in two variables b substitution: 1. Solve one of the equations for one of the variables.. Substitute the epression obtained in step 1 for that variable in the other equation, and solve the resulting equation in one variable.. Substitute the value for that variable into one of the original equations and solve for the other variable. a. if we get 0 = nonzero number or # = different # (contradiction), the sstem is inconsistent, the lines are parallel, the equations are independent, and there is no solution. b. if we get 0= 0 or # = same # (identit), the sstem is consistent, the lines are the same (coincide), the equations are dependent, and there are infinitel man solutions. Eample: Solve the sstem of linear equations using the substitution method. + 5 = = 8 This equation is solved for. ( ) = 1+ 5 = 7 = 7 = 1 = Substitute 8 in for in the first equation. Solve for. = 8 ( ) = Substitute in for to find the value of. The solution is (,). Check in both original equations: + 5 = = 8 ( ) + 5 ( ) = = = = 8 1 = = ( ) 55

60 Eample: SYSTEMS OF LINEAR EQUATIONS Solve the sstem of linear equations using the substitution method. + 1 = 5 + = 11 Since has a coefficient of 1, it will be eas to solve this second equation for. + 1 = = ( ) = Substitute in for = 6 = 18 = Solve for. ( ) = = 1 Substitute in for to find the value of. The solution is (,1). Check in both original equations: + 1 = ( ) ( ) = = = 5+ = 11 ( ) = = = 11 56

61 SYSTEMS OF LINEAR EQUATIONS Solving Linear Sstems B Elimination (Addition) Method To solve linear sstems of two equations in two variables b elimination: 1. Write the sstem so that each equation is in standard form. a + b = c. Multipl one equation (or both equations if necessar), b a number to obtain additive inverse (opposite) coefficients of one of the variables.. Add the resulting equations and solve the new equation in one variable.. Substitute the value for that variable into one of the original equations and solve for the other variable. a. If we get 0 = nonzero number or # = different # (contradiction), the sstem is inconsistent; there are no solutions, the equations are independent, and the lines are parallel. Eample: b. If we get 0 = 0 or # = same # (identit), the sstem is consistent; there are infinitel man solutions, the equations are dependent and the lines are the same (coincide). Solve the sstem of linear equations using the elimination (addition) method. + = 9 = = 6 = The coefficients of are opposites (additive inverses). Add these equations. Solve for. + = 9 + = 9 = 7 The solution is (,7). Substitute in for and solve for. Check in both original equations: + = 9 + 7= 9 9= 9 = ( ) 7= 7= = 57

62 Eample: SYSTEMS OF LINEAR EQUATIONS Solve this sstem of linear equations using the elimination (addition) method. + = 11 8 = Multipling the first equation b ields 8.This will result in opposite coefficients (additive inverses) of = 8 = 1 = = Add these equations. Solve for. ( ) + = = 11 = = 1 Substitute in for and solve for. The solution is (,1). Check in both original equations: + = 11 ( ) ( ) + 1 = = = 11 8 = ( ) ( ) 81 = 6 8= = 58

63 SYSTEMS OF LINEAR EQUATIONS Problem Solve this Sstem of Linear Equations using all three methods: Graphing Method Substitution Method Elimination (Addition) Method + = 1 = 1 Graphing Method: 0 Substitution Method: Elimination (Addition) Method: Answer is on the following page. 59

64 SYSTEMS OF LINEAR EQUATIONS Answer: 0 Solution: 60

65 SYSTEMS OF LINEAR EQUATIONS Problems Solve these sstems of linear equations using either the substitution or elimination (addition) method. Answers 1. + = 11 = 6 and = 5 1 6,5 = ( ). + = 9 = and = 1 11,1 = ( ). + = 5 = and = 1 5 8,1 = ( ). + = 7 = 1 and = ,1 = ( ) 5. + = 10 6 = 1 1 = and = 1 1, = = 1 and = 11 1, = ( ) 7. + = = 1 and = 8 1, + = ( ) = 5 = 1 and = 0 1, 0 + = ( ) 9. = 8 = = 6 = 1 Inconsistent sstem with no solution. Dependent sstem with infinitel man solutions. 61

66 APPLICATION OF A SYSTEM OF LINEAR EQUATIONS The following steps are helpful when solving problems involving two unknown quantities. 1. Analze the problem b reading it carefull to understand the given facts. Often a diagram or table will help ou visualize the facts of the problem.. Define variables to represent the two unknown quantities.. Translate the words of the problem to form two equations involving each of the two variables.. Solve the sstem of equations using graphing, substitution, or elimination (addition) method. 5. State the conclusion. 6. Check the results in the words of the problem. Eample: Determine the cost of a quart of pineapple and a container of frozen raspberries for the punch Diane is making. Three () quarts of pineapple juice and containers of raspberries will cost $10. Five (5) quarts of pineapple juice and containers of raspberries will cost $1. Let: p = cost of a quart of pineapple juice Define the variables r = cost of a container of raspberries p+ r = 10 5 p+ r = 1 Solve the sstem for p using elimination (addition) method p+ r = p r = 7 p = 1 p = ( ) + r = 10 r = 1 Substitute in for p and solve for r Answer: A quart of pineapples costs $ and a container of frozen raspberries costs $1. Check: Using $ as the cost for a quart of pineapple juice, quarts costs $6. Using $1 as the cost of a container of raspberries, containers cost $. $6 for pineapple juice and $ for raspberries is $10 total cost. Do the same technique for checking the $1 cost. Check the results in the words of the problem. 6

67 APPLICATION OF A SYSTEM OF LINEAR EQUATIONS Problem Answer Use the steps for solving a sstem of linear equations. People have begun purchasing tickets to a production of a musical at a regional theatre. A purchase of 9 adult tickets and 7 tickets for the children costs $116. Another purchase of 5 adult tickets and 8 tickets for the children costs $85. Find the cost of an adult ticket and a child s ticket. A child s ticket costs $5 and an adult ticket costs $9. 6

68 SOLVING LINEAR INEQUALITIES IN TWO VARIABLES Graphing Linear Inequalities In Two Variables 1. Replace the inequalit smbol with an equal smbol = and graph the boundar line of the region. If the original inequalit allows the possibilit of equalit (the smbol is either or ), draw the boundar line as a solid line. If equalit is not allowed (< or >), draw the boundar line as a dashed line.. Pick a test point that is on one side of the boundar line. (Use the origin if possible; it is easier). Replace and in the inequalit with the coordinates of that point. If a true statement results, shade the side (half-plane) that contains that point. If a false statement results, shade the other side (other half-plane). Eample: Solve + Step 1: + = = + sign, draw the solid boundar line Step : pick (0,0) ( ) 0+ 0 is true Pick test point. Shade the side that contains (0,0).. 0 ( 0,0) 6

69 SOLVING LINEAR INEQUALITIES IN TWO VARIABLES Problems 1 1. Solve 1. Solve + < 6. Solve 5 65

70 SOLVING LINEAR INEQUALITIES IN TWO VARIABLES Answers

71 SOLVING SYSTEMS OF LINEAR INEQUALITIES 1. Graph each inequalit on the same rectangular coordinate sstem.. Use shading to highlight the intersection of the graphs (the region where the graphs overlap). The points in this region are the solutions of the sstem.. As an informal check, pick a point from the region where the graphs intersect and verit that its coordinates satisf each inequalit of the original sstem. Eample: Solve this sstem of inequalities. + > 9 Look at the graph of each inequalit separatel. + sign, draw solid boundar line. Use (0, 0) as a test point. 0 is false. Shade the half-plane that does not include (0, 0). > 9 > sign, draw dashed boundar line. Use (0, 0) as a test point. 0 > 9 is false. Shade the half-plane that does not include (0, 0). 67

72 SOLVING SYSTEMS OF LINEAR INEQUALITIES + > 9 0 Graph each inequalit on the same coordinate sstem. Notice the region where the graphs overlap. The points in this region are the solutions of the sstem. Solution of the Sstem: 0 68

73 SOLVING SYSTEMS OF LINEAR INEQUALITIES Informal Check: The point (5,) is in the region of overlap. Substitute (5,) in to both inequalities is true > 9 5 ( ) ( ) > 9 is true 0 (5,) (5,) is one of the solutions. 69

74 SOLVING SYSTEMS OF LINEAR INEQUALITIES Problems 1. Solve this sstem of linear inequalities. + 5 > Solve this sstem of linear inequalities Answers on the net page 70

75 SOLVING SYSTEMS OF LINEAR INEQUALITIES Answers

76 SOLVING ABSOLUTE VALUE EQUATIONS Absolute Value: The absolute value of a number is its distance from 0 on the number line For an positive number k and an algebraic epression X: To solve X = k, solve the equivalent compound equation X = k or X = k Eample: = 7 Solution: =7 or = Eample: = = or = Solution: =5 = Check: = 5 = 5 = = = Check: = 1 = 1 = = = 7

77 SOLVING ABSOLUTE VALUE EQUATIONS Eample: Solve the following: 6 = Solve the equation b rewriting it as two separate equations. 6 6 = or = When X and X = k = k, then X = k Solve each equation for. 6 = 18 6 = 18 Multipl both sides b. Solutions: = 19 = 6 Add 6 to both sides. = 8 or = -16 Divide both sides b. Check: Check = 8 Check = 16 ( ) 8 6 = ( ) 16 6 = 19 6 = 6 6 = 18 = 18 = = = = = The two solutions check. 7

78 SOLVING ABSOLUTE VALUE EQUATIONS Eample: Solve the following: 5 7 = ( + 1) Solve the equation b rewriting it as two separate equations. ( ) ( ) 5 7= + 1 or 5 7= + 1 When X X = k, then = k and X = k Solve each equation for. Solutions: [ ] 5 7= + 5 7= + 7 = 5 7= = = 9 = = 11 or 1 = Use the Distributive Propert. Check: Check = 11 Check 1 = ( ) = ( + ) ( ) 55 7 = 1 8 = 8 8 = = = = = The two solutions check. 7

79 SOLVING ABSOLUTE VALUE EQUATIONS Problems Answers 1. = 5 7 = or = = 0 No solution. You can t solve an absolute value equation when the absolute value is equal to a negative quantit = 9 7 = or =. 1 5 = = = = 11 or = 6. 1 ( 1) + = + = or = 5 75

80 SOLVING ABSOLUTE VALUE INEQUALITIES To solve inequalities with absolute value signs, there are two cases: For an positive number k and an algebraic epression X Case 1: To solve X < k, solve the equivalent compound inequalit k < X < k. To solve X k, solve the equivalent compound inequalit k X k. Eample: Solve: < 7 Solution: -7<<7 Interval Notation: (-7,7) or For an positive number k and an algebraic epression X Case : To solve X > k, solve the equivalent compound inequalit X < k or X > k. To solve X k, solve the equivalent compound inequalit X k or X k. Eample: Solve: 7 Solution: 7 or 7 Interval Notation:, 7] [7, or

81 SOLVING ABSOLUTE VALUE INEQUALITIES Eample: Solve: Case 1 Equivalent compound inequalit 1 5 Addition Propert of Equalit Solution: 1 5 Interval Notation: [ 1,5] or Eample: Solve: > Case < or > Equivalent inequalities < 1 > 5 Interval Notation: (, 1) ( 5, ) Solution: < 1 or > or

82 SOLVING ABSOLUTE VALUE INEQUALITIES Problems Answers 1. 5 < < < 8,8 ( ). 7 > 10 < or > 17, 17, ( ) ( ) ,. < 9 < < 6, 6 ( ) or (, 7], ) or,, [ ) 78

83 VOCABULARY USED IN APPLICATION PROBLEMS Addition add English Algebra sum The sum of a number and. + total Seven more than a number. + 7 plus Si increased b a number. 6 + in all A number added to more than A number plus. + together increased b all together combined Subtraction subtracted from English Algebra difference The difference of a number and. take awa The difference of and a number. less than Five less than a number. 5 minus A number decreased b. remain A number subtracted from 8. 8 decreased b Eight subtracted from a number. 8 have left Two minus a number. are left more fewer Be careful with subtraction. The order is important. Three less than a number is. 79

84 VOCABULARY USED IN APPLICATION PROBLEMS Multiplication English Algebra product of The product of and a number. multiplied b Three-fourths of a number. times Four times a number. of A number multiplied b 6. 6 Double a number. Twice a number. Division English Algebra divided b quotient of separated into equal parts shared equall The quotient of a number and. The quotient of and a number. A number divided b 6. Si divided b a number. or or 6 or 6 or 6 6 Be careful with division. The order is important. A number divided b 6 is 6 80

85 SOLVING APPLICATION PROBLEMS Strateg for solving word problems: 1. Analze the problem: Read the problem carefull.. Visualize the facts of the problem (if needed): Use diagrams and/or tables.. Define the variable/s: Identif the unknown quantit (or quantities) and label them, i. e., let = something.. Write an equation: Use the defined variable/s. 5. Solve the equation: Make sure ou have answered the question that was asked. 6. Check our answer(s): Use the original words of the problem. Eamples: 1. The sum of three times a number and 11 is 1. Find the number. Let = the number + 11 = 1 Define the variable. Write an equation = 1 11 = = 8 Solve the equation. Solution: The number is 8. Check: ( ) 8 = + 11 = 1. Together, a lot and a house cost $0,000. The house costs seven times more than the lot. How much does the lot cost? The house? Let = the cost of the lot 7 = the cost of the house + 7= 0,000 8 = 0,000 = 5,000 7 = = 5,000 Solution: The lot costs $5,000. The house costs $5,000. Define the variable. Write an equation Solve the equation. Check: 5, ,000 = 0, = 5,000 81

86 SOLVING APPLICATION PROBLEMS Problems: 1. Five plus three more than a number is nineteen. What is the number?. When 18 is subtracted from si times a certain number, the result is 96. What is the number?. If ou double a number and then add 85, ou get three-fourths of the original number. What is the original number?. A 180-m rope is cut into three pieces. The second piece is twice as long as the first. The third piece is three times as long as the second. How long is each piece of rope? 5. Donna and Melissa purchased rollerblades for a total of $107. Donna paid $17 more for her rollerblades than Melissa did. What did Melissa pa? 6. A student pas $78 for a calculator and a keboard. If the calculator costs $6 less than the keboard, how much did each cost? 8

87 SOLVING APPLICATION PROBLEMS Answers ( ) = 19 The number is 11. = = 96 = 19 The number is = = 68 The number is 68.. ( ) + + = 180 = 0 The lengths are 0 m, 0 m, and 10 m = 107 = 5 Melissa paid $5. 6. ( ) + 6 = 78 = 171 The keboard costs $171, and the calculator costs $107. 8

88 SOLVING APPLICATION PROBLEMS Problems Solve the following word problems using one variable or two variables in a sstem. Integer Problems 1. The sum of three consecutive integers is 1. Find the integers.. The sum of three consecutive even integers is 8. Find the integers.. The sum of three consecutive odd integers is 111. Find the integers. Perimeter Problems. The length of a rectangle is seven more than the width. The perimeter is inches. Find the length and width. 5. The length of a rectangle is inches less than twice the width. The perimeter is 18 inches. Find the length and width. 6. The length of one side of a triangle is inches more than the shortest side. The longest side is two inches more than twice the length of the shortest side. The perimeter is 6 inches. Find the length of all three sides. 8

89 SOLVING APPLICATION PROBLEMS Miture Problems 7. For Valentine s Da, Cand, a cand store owner, wants a miture of cand hearts and foilwrapped chocolates. If she has 10 pounds of cand hearts, which sell for $ per pound, how man pounds of the chocolates, which sell for $6 per pound, should be mied to get a miture selling at $5 per pound? 8. The same cand store owner has pounds of chocolate creams which sell for $1 per pound. How man pounds of chocolate caramel nut clusters, which sell for $9 per pound, should he mi to get a miture selling at $10 per pound? 9. Sharon wants to make 100 pounds of holida mi for her baskets. She purchases cashews at $8.75 per pound and walnuts at $.75 per pound. She feels she can afford a miture which costs $6.5 per pound. How much of each tpe of nut should she purchase to make the mi? Distance Problems 10. Two cars leave Chicago at 11 a.m. headed in opposite directions. At p.m., the two cars are 75 miles apart. If one car is traveling 5 mph faster than the other, what are their speeds? 11. A plane leaves Chicago headed due west at 10 a.m. At 11 a.m., another plane leaves Chicago headed due east. At 1 p.m., the two planes are 950 miles apart. If the first plane is fling 100 mph slower than the second plane, find their rates. 1. A car leaves Milwaukee at noon headed north. Five hours later it arrives at its destination. A second car traveling south at a rate 10 mph slower leaves Milwaukee at :00 and arrives at its destination two hours later. When the arrive at their destination, the are 00 miles apart. How fast is each of the cars traveling? 1. A bicclist can ride miles with the wind in hours. Against the wind, the return trip takes him hours. Find the speed of the wind. (Hint: Solve using sstems of equations.) 85

90 SOLVING APPLICATION PROBLEMS Investment Problems 1. A couple wants to invest $1,000 in two retirement accounts, one earning 6% and the other 9%. How much should be invested in each account for them to earn an annual interest of $95? 15. An investor has put $,500 in a credit union account earning % annual interest. How much should he invest in an account which pas 10% annual interest to receive total annual interest of $1,000 from the two accounts. 16. Three accounts generate a total annual interest of $1,9.50. The investor deposited an equal amount of mone in each account. The accounts paid an annual rate of return of 7%, 8%, and 10.5%. How much was invested in each account? Number-Value Problems 17. The admission prices for a movie theater in Crstal Lake are $9 for adults, $8 for seniors, and $5 for children. A famil purchased twice as man children s tickets as adults and the same number of senior tickets as adults. The total cost of the tickets was $5. How man of each tpe of ticket was purchased? 18. Marie has $.0 worth of quarters, dimes, and nickels. She has times as man nickels as quarters and fewer dimes than quarters. How man of each tpe of coin does she have? 19. Deb went shopping at the school bookstore and purchased $5 of computer items. The CD s cost $ each, the DVD s cost $ each, and the flash drives were $15 each. The total cost was $5. He purchased one more DVD than CD s and half as man flash drives as CD s. How man of each did he purchase? 86

Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7

Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7 Practice Solving Sstems b Graphing Solve each sstem b graphing. Check our solution. 1. =- + 3 = - (1, ). = 1 - (, 1) =-3 + 5 3. = 3 + + = 1 (, 3). =-5 = - 7. = 3-5 3 - = 0 (1, 5) 5. -3 + = 5 =-7 (, 7).

More information

California State University, Northridge

California State University, Northridge California State Universit, Northridge MATH 09 HYBRID WORKBOOKS Spring 00 Chapter Equations, Inequalities and Applications. The Addition Propert of Equalit Learning Objectives:. Use the Addition Propert

More information

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression

Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important

More information

Course 15 Numbers and Their Properties

Course 15 Numbers and Their Properties Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.

More information

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 2nd edition. Miller, O'Neill, & Hyde. Victor Valley College

Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 2nd edition. Miller, O'Neill, & Hyde. Victor Valley College Lecture Guide Math 90 - Intermediate Algebra to accompan Intermediate Algebra, 2nd edition Miller, O'Neill, & Hde Prepared b Stephen Toner Victor Valle College Last updated: 11/24/10 0 1.1 Sets of Numbers

More information

math0320 FALL interactmath sections developmental mathematics sullivan 1e

math0320 FALL interactmath sections developmental mathematics sullivan 1e Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan

More information

Module 3, Section 4 Analytic Geometry II

Module 3, Section 4 Analytic Geometry II Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related

More information

Math 154A Elementary Algebra Fall 2014 Final Exam Study Guide

Math 154A Elementary Algebra Fall 2014 Final Exam Study Guide Math A Elementar Algebra Fall 0 Final Eam Stud Guide The eam is on Tuesda, December 6 th from 6:00pm 8:0pm. You are allowed a scientific calculator and a " b 6" inde card for notes. On our inde card be

More information

Name Date PD. Systems of Equations and Inequalities

Name Date PD. Systems of Equations and Inequalities Name Date PD Sstems of Equations and Inequalities Sstems of Equations Vocabular: A sstem of linear equations is A solution of a sstem of linear equations is Points of Intersection (POI) are the same thing

More information

Decimal Operations No Calculators!!! Directions: Perform the indicated operation. Show all work. Use extra paper if necessary.

Decimal Operations No Calculators!!! Directions: Perform the indicated operation. Show all work. Use extra paper if necessary. Decimal Operations No Calculators!!! Directions: Perform the indicated operation. Show all work. Use etra paper if necessar. Find.8 +.9...09 +. + 0.06 =. 6.08 + 6.8 + 00. =. 8. 6.09 =. 00.908. = Find.

More information

Chapter 1-2 Add and Subtract Integers

Chapter 1-2 Add and Subtract Integers Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign

More information

Algebra 2 Honors Summer Packet 2018

Algebra 2 Honors Summer Packet 2018 Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear

More information

Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.

Algebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions. Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) C) 31.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) C) 31. Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the sentence as a mathematical statement. 1) Negative twent-four is equal to negative

More information

Section 3.1 Solving Linear Systems by Graphing

Section 3.1 Solving Linear Systems by Graphing Section 3.1 Solving Linear Sstems b Graphing Name: Period: Objective(s): Solve a sstem of linear equations in two variables using graphing. Essential Question: Eplain how to tell from a graph of a sstem

More information

Intermediate Algebra Review for Exam 1 - Spring 2005

Intermediate Algebra Review for Exam 1 - Spring 2005 Intermediate Algebra Review for Eam - Spring 00 Use mathematical smbols to translate the phrase. ) a) 9 more than half of some number b) 0 less than a number c) 37 percent of some number Evaluate the epression.

More information

Eam Name algebra final eam review147 aam032020181t4highschool www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation.

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

REVIEW PACKET FOR END OF COURSE EXAM

REVIEW PACKET FOR END OF COURSE EXAM Math H REVIEW PACKET FOR END OF COURSE EXAM DO NOT WRITE ON PACKET! Do on binder paper, show support work. On this packet leave all fractional answers in improper fractional form (ecept where appropriate

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

More information

Diaz Math 080 Midterm Review: Modules A-F Page 1 of 7

Diaz Math 080 Midterm Review: Modules A-F Page 1 of 7 Diaz Math 080 Midterm Review: Modules A-F Page 1 of 7 1. Use the rule for order of operations to simplif the epression: 11 9 7. Perform the indicated operations and simplif: 7( + ) 6(5 9) 3. If a = 3,

More information

Unit 2 Notes Packet on Quadratic Functions and Factoring

Unit 2 Notes Packet on Quadratic Functions and Factoring Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a

More information

Math 0308 Final Exam Review(answers) Solve the given equations. 1. 3x 14 8x 1

Math 0308 Final Exam Review(answers) Solve the given equations. 1. 3x 14 8x 1 Math 8 Final Eam Review(answers) Solve the given equations.. 8.. 9.. 9 9 8 8.. 8 8 all real numbers 8. 9. all real numbers no solution 8 8 9 9 9 Solve the following inequalities. Graph our solution on

More information

Mt. Douglas Secondary

Mt. Douglas Secondary Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 57.1 Review: Graphing a Linear Equation A linear equation means the equation of a straight line, and can be written in one of two forms.

More information

Math Departmental Exit Assessment Review (Student Version)

Math Departmental Exit Assessment Review (Student Version) Math 008 - Departmental Eit Assessment Review (Student Version) Solve the equation. (Section.) ) ( + ) - 8 = 6-80 - 0 ) + - - 7 = 0-60 - 0 ) 8 + 9 = 9 - - ) - = 60 0-0 -60 ) 0.0 + 0.0(000 - ) = 0.0 0 6000

More information

Mourning Sr. High. Pre-Calculus 2017 Summer Assignment. 2. Write the final answer on the line provided.

Mourning Sr. High. Pre-Calculus 2017 Summer Assignment. 2. Write the final answer on the line provided. Mourning Sr. High Directions: 1. Show all of our work. Pre-Calculus 017 Summer Assignment. Write the final answer on the line provided. This assignment will be collected the first da of school. You will

More information

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.

Algebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology. Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph

More information

Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities

Ready To Go On? Skills Intervention 2-1 Solving Linear Equations and Inequalities A Read To Go n? Skills Intervention -1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular equation solution of an equation linear

More information

review for math TSI 182 practice aafm m

review for math TSI 182 practice aafm m Eam TSI 182 Name review for math TSI 182 practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif.

More information

Diaz Math 080 Midterm Review: Modules A-F Page 1 of 7

Diaz Math 080 Midterm Review: Modules A-F Page 1 of 7 Diaz Math 080 Midterm Review: Modules A-F Page 1 of 7 1. Use the rule for order of operations to simplif the epression: 11 9 7. Perform the indicated operations and simplif: 7(4 + 4) 6(5 9) 3. If a = 3,

More information

Rational Equations. You can use a rational function to model the intensity of sound.

Rational Equations. You can use a rational function to model the intensity of sound. UNIT Rational Equations You can use a rational function to model the intensit of sound. Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,

More information

PRE-ALGEBRA SUMMARY WHOLE NUMBERS

PRE-ALGEBRA SUMMARY WHOLE NUMBERS PRE-ALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in

More information

x. 4. 2x 10 4x. 10 x

x. 4. 2x 10 4x. 10 x CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain

More information

Study Guide for Math 095

Study Guide for Math 095 Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

More information

1. Simplify each expression and write all answers without negative exponents. for variable L.

1. Simplify each expression and write all answers without negative exponents. for variable L. MATH 0: PRACTICE FINAL Spring, 007 Chapter # :. Simplif each epression and write all answers without negative eponents. ( ab ) Ans. b 9 7a 6 Ans.. Solve each equation. 5( ) = 5 5 Ans. man solutions + 7

More information

NOTES. [Type the document subtitle] Math 0310

NOTES. [Type the document subtitle] Math 0310 NOTES [Type the document subtitle] Math 010 Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis.

More information

Summer Math Packet (revised 2017)

Summer Math Packet (revised 2017) Summer Math Packet (revised 07) In preparation for Honors Math III, we have prepared a packet of concepts that students should know how to do as these concepts have been taught in previous math classes.

More information

math FALL developmental mathematics sullivan 1e

math FALL developmental mathematics sullivan 1e TSIpractice eam review 1 131 180 plus 34 TSI questions for elementar and intermediate algebra m0300004301 aaa Name www.alvarezmathhelp.com math0300004301 FALL 01 100 interactmath developmental mathematics

More information

A. Simplifying Polynomial Expressions

A. Simplifying Polynomial Expressions A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +

More information

West Campus State Math Competency Test Info and Practice

West Campus State Math Competency Test Info and Practice West Campus State Math Competenc Test Info and Practice Question Page Skill A Simplif using order of operations (No grouping/no eponents) A Simplif using order of operations (With grouping and eponents)

More information

Diagnostic Tests Study Guide

Diagnostic Tests Study Guide California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra

More information

Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook

Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Identify the domain and the range of the relation from the graph. 8)

Identify the domain and the range of the relation from the graph. 8) INTERMEDIATE ALGEBRA REVIEW FOR TEST Use the given conditions to write an equation for the line. 1) a) Passing through (, -) and parallel to = - +. b) Passing through (, 7) and parallel to - 3 = 10 c)

More information

Math Intermediate Algebra

Math Intermediate Algebra Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and

More information

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution

More information

review math0410 (1-174) and math 0320 ( ) aafinm mg

review math0410 (1-174) and math 0320 ( ) aafinm mg Eam Name review math04 (1-174) and math 0320 (17-243) 03201700aafinm0424300 mg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif. 1) 7 2-3 A)

More information

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations

Ch 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4

More information

Intermediate Algebra 100A Final Exam Review Fall 2007

Intermediate Algebra 100A Final Exam Review Fall 2007 1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,

More information

C) x m A) 260 sq. m B) 26 sq. m C) 40 sq. m D) 364 sq. m. 7) x x - (6x + 24) = -4 A) 0 B) all real numbers C) 4 D) no solution

C) x m A) 260 sq. m B) 26 sq. m C) 40 sq. m D) 364 sq. m. 7) x x - (6x + 24) = -4 A) 0 B) all real numbers C) 4 D) no solution Sample Departmental Final - Math 46 Perform the indicated operation. Simplif if possible. 1) 7 - - 2-2 + 3 2 - A) + - 2 B) - + 4-2 C) + 4-2 D) - + - 2 Solve the problem. 2) The sum of a number and its

More information

Chapter 5: Systems of Equations

Chapter 5: Systems of Equations Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.

More information

13.1 2X2 Systems of Equations

13.1 2X2 Systems of Equations . X Sstems of Equations In this section we want to spend some time reviewing sstems of equations. Recall there are two basic techniques we use for solving a sstem of equations: Elimination and Substitution.

More information

Part I: No Calculators Allowed

Part I: No Calculators Allowed Name of Math lover: m favorite student Math 60: ELEMENTARY ALGEBRA Xtra Practice for the Last Eam Question : What s going to be on the final eam? Answer : Material from Chapter 1 to Chapter 9. Date: To

More information

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions

Ready To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte

More information

On a video game, Jacob got 1685 points and earned two bonuses worth 193 and 270 points. What is his total score? Answer: 2148 points

On a video game, Jacob got 1685 points and earned two bonuses worth 193 and 270 points. What is his total score? Answer: 2148 points Chapter Numerical Expressions and Factors Information Frame 9. Sample answers are given.. Ke Words: the sum of, the total of Real-Life Application : On a video game, Jacob got 68 points and earned two

More information

(c) ( 5) 2. (d) 3. (c) 3(5 7) 2 6(3) (d) (9 13) ( 3) Question 4. Multiply using the distributive property and collect like terms if possible.

(c) ( 5) 2. (d) 3. (c) 3(5 7) 2 6(3) (d) (9 13) ( 3) Question 4. Multiply using the distributive property and collect like terms if possible. Name: Chapter 1 Question 1. Evaluate the following epressions. (a) 5 (c) ( 5) (b) 5 (d) ( 1 ) 3 3 Question. Evaluate the following epressions. (a) 0 5() 3 4 (c) 3(5 7) 6(3) (b) 9 + (8 5) (d) (9 13) + 15

More information

Coached Instruction Supplement

Coached Instruction Supplement Practice Coach PLUS Coached Instruction Supplement Mathematics 8 Practice Coach PLUS, Coached Instruction Supplement, Mathematics, Grade 8 679NASP Triumph Learning Triumph Learning, LLC. All rights reserved.

More information

Lecture Guide. Math 50 - Elementary Algebra. Stephen Toner. Introductory Algebra, 2nd edition. Miller, O'Neil, Hyde. Victor Valley College

Lecture Guide. Math 50 - Elementary Algebra. Stephen Toner. Introductory Algebra, 2nd edition. Miller, O'Neil, Hyde. Victor Valley College Lecture Guide Math 50 - Elementar Algebra to accompan Introductor Algebra, 2nd edition Miller, O'Neil, Hde Prepared b Stephen Toner Victor Valle College Last updated: 12/27/10 1 1.1 - Sets of Numbers and

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

Algebra 1 Skills Needed to be Successful in Algebra 2

Algebra 1 Skills Needed to be Successful in Algebra 2 Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed

More information

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs

Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The

More information

Northwest High School s Algebra 2/Honors Algebra 2

Northwest High School s Algebra 2/Honors Algebra 2 Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #11 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

Geometry 21 Summer Work Packet Review and Study Guide

Geometry 21 Summer Work Packet Review and Study Guide Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

More information

7.1 Solving Linear Systems by Graphing

7.1 Solving Linear Systems by Graphing 7.1 Solving Linear Sstems b Graphing Objectives: Learn how to solve a sstem of linear equations b graphing Learn how to model a real-life situation using a sstem of linear equations With an equation, an

More information

Honors Algebra

Honors Algebra Honors Algebra 08-09 Honors Algebra is a rigorous course that requires the use of Algebra skills. The summer work is designed to maintain and reinforce these prerequisite skills so as to prepare ou for

More information

One of your primary goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan.

One of your primary goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan. PROBLEM SOLVING One of our primar goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan. Step Step Step Step Understand the problem. Read the problem

More information

Algebra 2 CPA Summer Assignment 2018

Algebra 2 CPA Summer Assignment 2018 Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou

More information

Intermediate Algebra Math 097. Evaluates/Practice Tests. For solutions, refer to the back of the PAN.

Intermediate Algebra Math 097. Evaluates/Practice Tests. For solutions, refer to the back of the PAN. Intermediate Algebra Math 097 Evaluates/Practice Tests For solutions, refer to the back of the PAN. Page of 8 Take this practice test to be sure that ou are prepared for the final quiz in Evaluate.. Solve

More information

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Math 1 Final Eam Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve. 1) - - 6 = - + 7 1) ) 6 + 7(- - ) = -8 - ) ) - t + t = 6 t + 1 ) Solve

More information

An equation is a statement that states that two expressions are equal. For example:

An equation is a statement that states that two expressions are equal. For example: Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the

More information

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1 Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab

More information

Review Topics for MATH 1400 Elements of Calculus Table of Contents

Review Topics for MATH 1400 Elements of Calculus Table of Contents Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical

More information

Algebra One Dictionary

Algebra One Dictionary Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.

More information

Algebra 2 Unit 1 Practice

Algebra 2 Unit 1 Practice Algebra Unit Practice LESSON - Use this information for Items. Aaron has $ to rent a bike in the cit. It costs $ per hour to rent a bike. The additional fee for a helmet is $ for the entire ride.. Write

More information

Algebra 2 Semester Exam Review

Algebra 2 Semester Exam Review Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation

More information

Graph the linear system and estimate the solution. Then check the solution algebraically.

Graph the linear system and estimate the solution. Then check the solution algebraically. (Chapters and ) A. Linear Sstems (pp. 6 0). Solve a Sstem b Graphing Vocabular Solution For a sstem of linear equations in two variables, an ordered pair (x, ) that satisfies each equation. Consistent

More information

Ready To Go On? Skills Intervention 12-1 Inverse Variation

Ready To Go On? Skills Intervention 12-1 Inverse Variation 12A Find this vocabular word in Lesson 12-1 and the Multilingual Glossar. Identifing Inverse Variation Tell whether the relationship is an inverse variation. Eplain. A. Read To Go On? Skills Intervention

More information

Lecture Guide. Math 42 - Elementary Algebra. Stephen Toner. Introductory Algebra, 3rd edition. Miller, O'Neill, Hyde. Victor Valley College

Lecture Guide. Math 42 - Elementary Algebra. Stephen Toner. Introductory Algebra, 3rd edition. Miller, O'Neill, Hyde. Victor Valley College Lecture Guide Math 42 - Elementar Algebra to accompan Introductor Algebra, 3rd edition Miller, O'Neill, Hde Prepared b Stephen Toner Victor Valle College Accompaning videos can be found at www.mathvideos.net.

More information

Math 100 Final Exam Review

Math 100 Final Exam Review Math 0 Final Eam Review Name The problems included in this review involve the important concepts covered this semester. Work in groups of 4. If our group gets stuck on a problem, let our instructor know.

More information

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.

8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1. 8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,

More information

Math Analysis/Honors Math Analysis Summer Assignment

Math Analysis/Honors Math Analysis Summer Assignment Math Analysis/Honors Math Analysis Summer Assignment To be successful in Math Analysis or Honors Math Analysis, a full understanding of the topics listed below is required prior to the school year. To

More information

Algebra Semester 1 Final Exam Review Name: Hour: Date:

Algebra Semester 1 Final Exam Review Name: Hour: Date: Algebra Semester Final Exam Review Name: Hour: Date: CHAPTER Learning Target: I can appl the order of operations to evaluate expressions. Simplif the following expressions (combine like terms). x + x +

More information

Lesson 1C ~ The Four Operations

Lesson 1C ~ The Four Operations Lesson C ~ The Four Operations Find the value of each expression. Show all work.. 8 + 7 3. 3 0 3. + 3 8 6 + 8 8. 7 + 6 + 0. Mike hosted a fundraiser to benefit the Children s Hospital and the Cancer Societ.

More information

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

Bishop Kelley High School Summer Math Program Course: Algebra 2 A 06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems

More information

Algebra 2 Summer Work Packet Review and Study Guide

Algebra 2 Summer Work Packet Review and Study Guide Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the

More information

MEP Pupil Text 16. The following statements illustrate the meaning of each of them.

MEP Pupil Text 16. The following statements illustrate the meaning of each of them. MEP Pupil Tet Inequalities. Inequalities on a Number Line An inequalit involves one of the four smbols >,, < or. The following statements illustrate the meaning of each of them. > : is greater than. :

More information

Algebra 1 CP Semester Exam Review

Algebra 1 CP Semester Exam Review Name: Hr: Algebra CP Semester Eam Review GET ORGANIZED. Successful studing begins with being organized. Bring this packet with ou to class ever da. DO NOT FALL BEHIND. Do the problems that are assigned

More information

MATH 103 Sample Final Exam Review

MATH 103 Sample Final Exam Review MATH 0 Sample Final Eam Review This review is a collection of sample questions used b instructors of this course at Missouri State Universit. It contains a sampling of problems representing the material

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

June If you want, you may scan your assignment and convert it to a.pdf file and it to me.

June If you want, you may scan your assignment and convert it to a.pdf file and  it to me. Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular

More information

LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II

LESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS

More information

MATH 110: FINAL EXAM REVIEW

MATH 110: FINAL EXAM REVIEW MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set

More information

MATH 080 Final-Exam Review

MATH 080 Final-Exam Review MATH 080 Final-Exam Review Can you simplify an expression using the order of operations? 1) Simplify 32(11-8) - 18 3 2-3 2) Simplify 5-3 3-3 6 + 3 A) 5 9 B) 19 9 C) - 25 9 D) 25 9 Can you evaluate an algebraic

More information

H.Algebra 2 Summer Review Packet

H.Algebra 2 Summer Review Packet H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,

More information

Unit 26 Solving Inequalities Inequalities on a Number Line Solution of Linear Inequalities (Inequations)

Unit 26 Solving Inequalities Inequalities on a Number Line Solution of Linear Inequalities (Inequations) UNIT Solving Inequalities: Student Tet Contents STRAND G: Algebra Unit Solving Inequalities Student Tet Contents Section. Inequalities on a Number Line. of Linear Inequalities (Inequations). Inequalities

More information

Math 101, Basic Algebra. Solving Linear Equations and Inequalities

Math 101, Basic Algebra. Solving Linear Equations and Inequalities Math 101, Basic Algebra Author: Debra Griffin Name Chapter 2 Solving Linear Equations and Inequalities 2.1 Simplifying Algebraic Expressions 2 Terms, coefficients, like terms, combining like terms, simplifying

More information