1.1 Variables and Expressions How can a verbal expression be translated to an algebraic expression?

Transcription

1 1.1 Variables and Expressions How can a verbal expression be translated to an algebraic expression? Recall: Variable: Algebraic Expression: Examples of Algebraic Expressions: Different ways to show multiplication: Product Factors Powers Power: Base: Exponent: Symbols Words Meaning to the first power to the second power b 6 x n x to the nth power Evaluate: 1. Evaluate Evaluate 4 3

2 3. Evaluate Evaluate 2 4 Translating Expressions: You will often need to translate between verbal expressions and algebraic expressions. What is the difference between a verbal expression and an algebraic expression? Writing Algebraic Expressions **The phrase that always gives the most problems is less than Which expression means three less than seven? 7 3 or 3-7 This often gets confused with three less seven, which means Write an algebraic expression for each verbal expression. 1. Eight more than a number 2. 7 less the product of 4 and a number x 3. One third of the original area a 4. The product of 7 and m to the fifth power Write an algebraic expression for each verbal expression less than a number 6. 9 plus the product of 2 and the number d 7. Two thirds of the original volume V 8. The product of 3 and a to the seventh power

3 Writing Verbal Expressions **There are many ways to write verbal expressions that all mean the same thing Write a verbal expression for each algebraic expression. 1. 4m 3 2. c d 3. x 4 y 9 Write a verbal expression for each algebraic expression. 4. 7x y 5 16y 6. 8x 2 5 Practice Problems Evaluate each expression Write an algebraic expression for each verbal expression. 7. The sum of a number and less than a number t less than 3 times a number minus the quotient of r and Two-fifths of a number j squared 12. n cubed increased by 5

4 13. X more than A number less One-third of a number 16. F divided by The quotient of 45 and r increased by twice a number decreased by 3 times d 20. K squared minus divided by t to the fifth power 22. The area of a circle is the number π times the square of the radius. Write an expression that represents the area of a circle with radius r. Write a verbal expression for each algebraic expression y 24. w x r a n 3 p n3 30. a 2 18b m z2 5

5 1.2 Order of Operations How do you evaluate numerical and algebraic expressions using the Order of Operations? Order of Operations Step 1: Step 2: Step 3: Step 4: Evaluate Numerical Expressions Evaluate each expression using the order of operations Evaluate Numerical Expressions with Grouping Symbols Examples of Grouping Symbols: Evaluate each expression. 1. 2(5) + 3(4 + 3) 2. 2[5 + (30 6) 2 ]

6 (8 3) 3(3 + 2) Evaluate each expression. 5. (15 9) [(1 + 1) 3 4] (3+7) 8. 4[12 (6 2)] Evaluate Algebraic Expressions To evaluate an algebraic expression, replace the variables with their values. Then use the order of operations. 1. Evaluate a 2 (b 3 4c) if a = 7, b = 3, and c = 5 2. Evaluate x(y 3 + 8) 12 if x = 3 and y = 4 3. Evaluate 2(x 2 y) + z 2 if x = 4, y = 3, and z = 2

7 Real World Practice: 1. The Pyramid Arena in Memphis is the third largest pyramid in the world. The area of its base is 360,000 square feet, and it is 321 feet high. The volume of a pyramid is onethird of the product of the area of the base B and its height h. a. Write an expression that represents the volume of a pyramid. b. Find the volume of the Pyramid Arena in Memphis 2. According to market research, the average consumer spends $78 per trip to the mall on weekends and only $67 per trip during the week. a. Write an algebraic expression to represent how much the average consumer spends at the mall in x weekend trips and y weekday trips. b. Evaluate the expression to find what the average consumer spends after going to the mall twice during the week and 5 times on the weekends. Practice Problems: Evaluate each expression [3( )] [(6 3 9) 23] (11 7) (12 6) (1+6) (6) ( )

8 Evaluate each expression if a = 4, b = 6, and c = b a 15. 2a + (b 2 3) 16. b(9 c) a 2 Evaluate each expression if r = 2, s = 3, and t = r + 6t rs 19. (2t + 3r) s 2 + (r 3 8)5 21. t 2 + 8st + r r(r + s) A sales representative receives an annual salary s, an average commission each month c, and a bonus b for each sales goal that she reaches. a. Write an algebraic expression to represent her total earnings in one year if she receives four equal bonuses. b. Suppose her annual salary is $52,000 and her average commission is $1225 per month. If each bonus is $1150, how much does she earn in a year?

9 1.3 Open Sentences How do you solve open sentence equations and inequalities? Activator: Find the truth values of the following sentences: a. There are 12 inches in a foot. b. October is the 11 th month of the year. c = d. Broccoli tastes good. Open Sentence: Variable: If the open sentence has = then it is called an If the open sentence has <, >,, then it is called an a. It s the best movie of the year. b. She likes her job. c. x + 5 = 25 d. Which statement in the activator is an open sentence? Why? Replacement Set: Element: Solution Set: 1. 2a + 5 = 21 If the replacement set is {6, 7, 8, 9}, find the solution set. a 2a + 5 = 21 True or False? So the solution set is:

10 y 13 If the replacement set is {3, 4, 5, 6}, find the solution set. y 24 2y 13 True or False? So the solution set is: 3. 6n + 7 = 37 If the replacement set is {3, 4, 5, 6, 7}, find the solution set. n 6n + 7 = 37 True or False? So the solution set is: > 2y 5 If the replacement set is {5, 6, 7, 8}, find the solution set. y 19 > 2y 5 True or False? So the solution set is:

11 Practice Problems: Find the solution of each equation is the replacement sets are a = {0, 3, 5, 8, 10} and b = {12, 17, 18, 21, 25} 1. b 12 = = 34 b a = (a 1) = a 4 = = a Find the solution set for each inequality using the given replacement set 7. s 2 < 6 {6, 7, 8, 9, 10, 11} 8. 5a + 7 > 22 {3, 4, 5, 6, 7} m {1, 3, 5, 7, 9, 11} 10. 2a 4 8 {12, 14, 16, 18, 20, 22} (x + 5) {1.2, 1.3, 1.4, 1.5} 12. x { 1, 1, 3, 1, 4, 11 } 4 2 4

12 1.4, 1.5, 1.6 Properties of Algebra What are the properties of Algebra, and how can you recognize these properties? Use your flip book to help you recall the properties below. Additive Identity Multiplicative Identity Additive Inverse Multiplicative Inverse Substitution Symmetric Property transitive Property commutative Property (addition) Commutative property (multiply) associative property (addition) Name the property shown by each statement = (a 6) 5 = a (6 5) = 12 Name the property shown by each statement = (2 + 5) + m = 2 + (5 + m) = 17 Find the value of n in each equation. Then name the property that is used n = n = y = 5 4. n 9 = x = 0 6. n 1 5 = 1

13 Simplifying Expressions Term: Coefficient: Like Terms: Equivalent Expressions: Simplest Form: Combine Like Terms to Simplify. If already in simplest form, write simplified x + 18x 2. 3n n + 9n p 2 8p 2p 2 Combine Like Terms to Simplify. If already in simplest form, write simplified a + 21a 5. 12x 2 8x 2 + 6x 6. b b + 13 Distributive Property What does the word distribute mean? Give an example. In math, distribute means to multiply a value by all other quantities in parentheses. The Distributive Property: This can be used for numerical and algebraic expressions.

14 Rewrite each product using the Distributive Property, then simplify. 1. 5(g 9) 2. 3(x 2 + x 1) 3. 6(r s t) Rewrite each product using the Distributive Property, then simplify (y + 3) 5. (8 + n)2 6. 4(x 2 + 8x + 2) Evaluate Expressions Using Multiple Properties Simplify the expression and name the property used in each step. 1. 6(x 2y) + 4( 3x + y) 2. Three times the sum of 3x and 2y added to five times the sum of x and 4y Simplify the expression and name the property used in each step. 3. 4(a + b) + 2(a + 2b) 4. 5 times the difference of q squared and r, plus eight times the sum of 3q and 2r

15 Practice Problems Find the value of n in each equation. Then name the property that is used 1. 1 = 2n 4. 4 n = = 6 + n = n 3. n 1 = 5 Rewrite each product using the Distributive Property, then simplify. 6. 2(x + 4) ( 1 2b) 3 7. (5 + n)3 8. 8(4 3m) 9. 3(x 6) 11. 4(p + q r) 12. 6(2 x 2 + x) 13. 5(6m 3 + 4n 3n) Simplify each expression. If not possible, write simplified x + 5y + 6x 19. 3(4m + n) + 2m(4 + n) 15. 5a + 3b + 2a + 7b 20. 6(0.4f + 0.2g) + 0.5f 16. 3(4x + 2) + 2x 21. x 2 + 3x + 2x + 5x m + 6n (d + 5)8 + 2f 18. 7(ac + 2b) + 2ac a b Simplify the expression and name the property used in each step (5) (3 2 5) Twice the sum of s and t, decreased by times the product of x and y increased by 3xy times the sum of x and y squared, minus 3 times the sum of x and y squared

16 1.8 Number Systems What are the relationships among the various number sets in the real number system? How do you order real numbers? Sets of Numbers Definition Examples Symbol Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers N W Z Q I R Classify Real Numbers Name ALL sets of numbers to which each real number belongs Name ALL sets of numbers to which each real number belongs

17 Closure Property If something is closed, what does that mean? Sets of numbers can also be closed, under certain operations. This means that if you perform that operation on numbers from the set, that the answer will also be from the set. 1. Is the set of whole numbers closed under multiplication? 2. Is the set of integers closed under division? 3. Is the set of real numbers closed under subtraction? 4. Is the set of whole numbers closed under subtraction? 5. Is the set of integers closed under addition? 6. Is the set of natural numbers closed under division? Comparing Real Numbers There are three signs we use to compare numbers. < > = Replace each with <, >, or = to make the sentence true Replace each with <, >, or = to make the sentence true

18 Ordering Real Numbers We can take this comparison a step further and compare multiple numbers at once. This is called ordering. 1. Order from least to greatest , 7, 8, Order from greatest to least , 0.2, 2, Order from least to greatest. 12 5, 6, 2. 4, Order from greatest to least. 0.42, 0. 63, 4 9 Practice Problems Name ALL sets of numbers to which each real number belongs

19 Determine whether each set of numbers is closed under the indicated operation. 9. Whole numbers, Division 14. Natural Numbers, Addition 10. Rational Numbers, Addition 11. Rational Numbers, Division 12. Natural Numbers, Subtraction 15. Whole Numbers, Multiplication 16. Integers, Subtraction 17. Integers, Multiplication 13. Irrational Numbers, Addition Replace each with <, >, or = to make the sentence true Order each set of numbers from least to greatest , 1 4, 0. 15, , 5 4 9, 13, , 16 49, , 0. 24, 9 144