Intermediate Algebra 100A Final Exam Review Fall 2007
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1 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, irrational numbers. List each set in roster form. List each set in set builder notation. 1. { < 4 and is an integer.}. {.6 < < 7 and is a whole number.} Consider A = {1, 4, 9}, and B = {1,,, 4,, 6}. Find A B 6. Find A B. Real numbers less than. 4. {0, 1,,, 4, } Consider the set { 1, 7,, 1, 0, 101, 1.}. 7. List the elements of the set that are integers. 8. List the elements of the set that are rational. 1. Properties of Real Numbers Words/Concepts to Know: absolute value, commutative property, associative property, identity property, inverse property, distributive property. ( ) Evaluate. 1.. ( ) 1. 6 ( ). 1 ( 9) 4. ( 1)( 9) Order of Operations Words/Concepts to Know: order of operations. Evaluate ( ). (6 9) (9 6) Eponents Product Rule for Eponents Quotient Rule for Eponents Negative Eponent Rule Raising a Power to a Power (the Power Rule) Zero Eponent Rule Raising a Product to a Power Raising a Quotient to a Power Simplify each epression and write the answer without 1. 1 y 8 y. ( y 1 y ) 1 a m a n = a m+n a m a = a m n n a m = 1 a m (a m ) n = a m n a 0 = 1 when a 0 (ab) m = a m b ( m a ) m b = a m b, and ( ) a m ( m b = b a negative eponents. ( 18m n p ). 6m n 4 p 4. ( 4 y 6y 8 ) 0 ) m Equations and Inequalities.1 Solving Linear Equations Words/Concepts to Know: terms, coefficient, degree of a term, like terms. State the degree of each term: 1. y. 6 y
2 Solve each equation. ( + ) = ( 4) 4. k 4 + = ( ) = (6 9) 6. 1 (6 ) = 1 ( 4). Problem Solving and Using Formulas Words/Concepts to Know: conditional equation, contradiction, identity. Solve each equation for the indicated variable y = 8, for y. y = 4, for y. 1 (y + 4) = 1 ( 6), for y 4. P = l + w, for w. Betsy is planning to build a rectangular sandbo for her daughter. She has 8 feet of wood to use for the sides. If the length is to be 11 feet, what is the width to be?. Applications of Algebra Phrase increased by more than sum of decreased by less than difference of twice times one-ninth of What it means in math Addition Subtraction Multiplication Division 1. Angles A and B are complementary angles(sum to 90 ). Determine the measures of A and B is angle A is 6 less than twice angle B.. Angles C and D are supplementary angles(sum to 180 ). Determine the measures of C and D if angle C is twice the difference of angle D and 1.. The sum of the measures of the angles of a triangle is 180. Find the measures of the three angles of the triangle if the largest angle is 4 more than the smallest angle and the third angle is more than the smallest angle..4 Additional Application Problems Words/Concepts to Know: motion formula, distance formula. 1. Dog Kennel A charges $7 plus $0.10 for each pound of the dog s weight to kennel animals per day. Dog Kennel B charges $16 per day to kennel dogs of any weight. Determine at which weight class the two kennels cost the same per day.. Two planes leave an airport at the same time traveling in opposite directions. The southbound plane travels at 0 miles per hour while the northbound plane travels at 40 miles per hour. After how many hours are the planes 490 miles apart?. Solving Linear Inequalities Words/Concepts to Know: compound inequality, intersection (and), union (or), number line, interval notation. Solve each inequality and write your solution on a number line Solve each inequality and write your solution in interval notation. 1. ( + 4) < ( 1) +. + < and or < 9 Page
3 Properties Used to Solve Inequalities If a > b, then a + c > b + c. If a > b, then a c > b c. If a > b, and c > 0, then ac > bc. If a > b, and c > 0, then a c > b c. If a > b, and c < 0, then ac < bc. If a > b, and c < 0, then a c < b c..6 Solving Equations and Inequalities Containing Absolute Values Be sure you can solve equations of the form: = a, < a, >. a, and = y. Find the solution set to the following equations. 1. =. 4 = Find the solution set to the following inequalities... 1 < Graphs and Functions.1 Graphs > = 7 Words/Concepts to Know: cartesian coordinate, y-ais, -ais, origin, ordered pair, four quadrants, graph, linear equation. Be sure you are able to identify objects on a graph, e.g. origin, -ais, y-intercept etc. Get in the habit of labeling your graphs, including all important points. 1. Are either of ( 1, ) or (1, ) solutions to the equation y = + 1?. Are either of (, ) or (1, ) solutions to the equation y = + 1?. Functions Words/Concepts to Know: dependent variable, independent variable, relation, function, domain, range, vertical line test, function notation. For the relations below find the domain and range, indicate See problems 1-40 on page 168 of your book for whether the relation is a function. etra practice. 1. {(1, 1), (1, ), (1, ), (1, 4)} Evaluate each of the following functions at f(0), f( ), f(1).. f() = 1 +. {(1, 1), (, 1), (, 1), (4, 1)}. 4. f() = +. Linear Functions: Graphs and Applications Words/Concepts to Know: linear function, standard form of a linear equation, -intercept, y-intercept, constant function. 1. Write the equations y = + 6 and 1 = y 7 in standard form.. Find the - and y- intercepts for the equations below. Try graphing the equations using the - and y- intercepts. y 4 = 1 f() = y = 60 = y = 4 Page
4 .4 The Slope-Intercept Form of a Linear Equation Words/Concepts to Know: slope of a line, positive/negative slope, zero slope, undefined, slope-intercept form. 1. Find the slope of the line through the given points. If the slope is undefined, so state. (, ) and (, 4) (, ) and ( 1, ) (, 7) and (, 16). Write each equation in slope-intercept form. Use the slope and y-intercept to graph each equation. + y = 9 y + = 4 + y = 10 =. The Point-Slope Form of a Linear Equation Words/Concepts to Know: point-slope form, parallel, perpendicular. 1. Write an equation in slope-intercept form for each line described below. m = through (1, ) m = 1 through (, ) m = 1 through ( 4, ). Write an equation in slope-intercept form for each line described below. (a) Through (4, 1) and perpendicular to the graph of y = + 1. (b) Through (, ) and perpendicular to the graph of + y =.. Determine whether the graphs of the two lines are parallel, perpendicular, or neither: y = 6 and + 6 = y.6 The Algebra of functions Words/Concepts to Know: operations on functions. 1. If f() = + 1 and g() = + find: (f + g)() (f g)() (f g)( 1) ( ) f () g 4 Systems of Equations and Inequalities 4.1 Solving Systems of Linear Equations in Two Variables Words/Concepts to Know: system of linear equations, consistent system, inconsistent system, dependent system, what does a solution represent on a graph, methods of solving a system of linear equations: graphing, substitution, addition. 1. Solve each system by substitution y = y = 4 +. Solve each system by addition. + y = 4 + y = 4 a + b = 4a b = y = y = 1 y = + y = Polynomials and Polynomial Functions.1 Addition and Subtraction of Polynomials Words/Concepts to Know: terms, polynomial, degree of a polynomial, leading term, leading coefficient, monomial, binomial, trinomial linear, quadratic, cubic, polynomial function, adding and subtracting polynomials. 1. Perform the indicated operation: ( + ) ( + 4) y ( y + 4y ) Page 4
5 . Multiplication of Polynomials Words/Concepts to Know: distributive property, FOIL, square a binomial, product of the sum and difference of two terms.. 1. Perform the indicated operation: ( + y y )( + 4y) ( ) ( + ) ( + )( 1).4 Factoring a Monomial from a Polynomial and Factoring by Grouping Words/Concepts to Know: greatest common factor (GCF), factoring, factor by grouping. 1. Factor: (a + ) + 1(a + ) (z + 4)(z + ) + (z 1)(z + ) 1 + 9y 4y y c c 4 + c c 4 ( ) 6( ) + 4( ) r 4 r 7r + 7r. Factoring Trinomials Words/Concepts to Know: methods for factoring trinomials, prime polynomial, factoring using substitution. 1. Factor the following trinomials completely: Special Factoring Formulas 1. Factor the following completely: a 4 4b 4 11y 4 49 a 4 + a b + b y.8 Polynomial Equations Words/Concepts to Know: polynomial equations, quadratic equation, zero-factor property, pythagorean theorem. 1. Solve: (a) = 0 (b) + 1 = = 0 4y 6y = 0 (e) y = 4y (f) y = y Rational Epressions and Equations 6.1 The Domains of Rational Functions and Multiplication and Division of Rational Epressions Words/Concepts to Know: rational epression, rational function, domain. 1. Determine the domain of each function. f() = f() = Multiply or divide as indicated. 4ab 4 6a b a b a b 6a + b a 9b a + ab a + a b Page
6 6. Addition and Subtraction of Rational Epressions Words/Concepts to Know: least common denominator (LCD). 1. Add or subtract as indicated. (a) (b) m m + 7m m + m 1 y 6 + y 6y y 4 9y 6.4 Solving Rational Equations 1. Solve for. (a) + = 1 (b) 1 4 = 4 = = Roots, Radicals, and Comple Numbers 7.1 Roots and Radicals Words/Concepts to Know: root, radical, principle square root, cube root, even inde, odd inde. 1. Evaluate 8 (a) 7 (b) 11. Write as an absolute value ( y) ( 4 + 1) 7. Rational Eponents 1. Evaluate (a) 6 1 (b) ( 81 ) 1. Write in eponential form. 7 ( 11 4 ( 64 1 ) 1 ) 1 ( 4 y) Simplifying Radicals Words/Concepts to Know: perfect square, perfect cube. 1. Simplify. (a) 48 (b) (e) 7 y 1 (f) 4 1 y 0 z 7 (g) (h) 16 4 y 10 00a 10 b 11 ab 4 Page 6
7 7.4 Adding, Subtracting, and Multiplying Radicals Words/Concepts to Know: like radicals. 1. Simplify. (a) (b) 4 + y( 7y y) ( 6 + )(4 6 1) 7.6 Solving Radical Equations 1. Solve. (a) = 4 (b) 4 + = = = 1 + (e) + 7 = 6 (f) 4 = Comple Numbers Words/Concepts to Know: imaginary unit (i), imaginary number, comple number, conjugate of a comple number, powers of i. 1. Solve. (a) ( 9 ) ( + 6 ) (b) ( + i)( i) i + i 18 (e) (f) Quadratic Equations 8.1 Solving Quadratic Equations by Completing the Square Words/Concepts to Know: square root property, perfect square trinomial, completing the square. 1. Use the square root property to solve each equation. 81 = 0 + = 0 ( + 6) 49 = 0. Solve each equation by completing the square. + = 0 10 = 6 = Solving Quadratic Equations by the Quadratic Formula Words/Concepts to Know: standard form of quadratic equation, quadratic formula, discriminant. 1. Use the discriminant to determine how many real solutions each equation has, then solve using the quadratic formula. (a) + = 0 (b) 6 = = 0 8. Graphing Quadratic Functions Words/Concepts to Know: verte, ais of symmetry, y-intercept, -intercept. 1. Determine whether the parabola opens upward or downward. Find the y-intercept. Find the verte. Find the intercepts (if any). Draw a graph. (a) f() = (b) y = f() = Page 7
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