Download Solving Equations

1-4
Solving Equations
Vocabulary
Review
Circle the equations.
2(3x 1 7) 5 10
3x 1 14
3x 1 2y 5 6
4x 1 (y 1 5)
Vocabulary Builder
x22x
inverse operations (noun)
IN
vurs ahp uh RAY shunz
inverse operations
Related Words: opposite, reverse
Main Idea: Inverse operations undo each other.
Use Your Vocabulary
1. Write each inverse operation.
Add 7.
Subtract 4.
Divide by 5.
Key Concepts Properties of Equality
Assume a, b, and c represent real numbers.
Property
Definition
Example
Reflexive
aa
55
Symmetric
If a b, then b a.
(4)(2) 8, so 8 (4)(2)
If a b and b c, then a c.
6 (2)(3) and (2)(3) (3)(2),
so 6 (3)(2)
If a b, then you can replace
a with b and vice versa.
If a b and 9 a 15,
then 9 b 15
Transitive
Substitution
Chapter 1
14
Multiply by 3.
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Example: Addition and subtraction are inverse operations.
Draw a line from each example in Column A to the property that it illustrates
in Column B.
Column A
Column B
2. a 5 4 and a 1 b 5 5, so 4 1 b 5 5
Reflexive Property
3. 2 1 (x 1 8) 5 2 1 (x 1 8)
Symmetric Property
4. x 1 y 5 z and z 5 4, so x 1 y 5 4
Transitive Property
5. y 1 12 5 9, so 9 5 y 1 12
Substitution Property
Key Concepts Properties of Equality, Continued
Assume a, b, and c represent real numbers.
Property
Definition
Example
Addition
If a b, then a c b c.
If x 12, then x 3 12 3.
Subtraction
If a b, then a c b c.
If x 12, then x 3 12 3.
Multiplication
If a b, then a ∙ c b ∙ c.
If x 12, then x ∙ 3 12 ∙ 3.
Division
If a b, then a c b c
(with c 0).
If x 12, then x 3 12 3.
Write the Property of Equality that justifies each statement.
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6. If x 1 2 5 5, then x 1 2 2 2 5 5 2 2.
Property of Equality
4x
7. If 4x 5 12, then 4 5 12
4.
x
Property of Equality
x
8. If 220 5 5 then 220 ? 5 5 5 ? 5.
Property of Equality
9. If y 2 3 5 212 then y 2 3 1 3 5 212 1 3.
Property of Equality
Problem 1 Solving a One-Step Equation
Got It? What is the solution of 12b 5 18?
10. Circle the multiplicative inverse of 12.
1
12
212
1
212
1
11. Use the multiplicative inverse to solve the equation and check your solution.
15
Lesson 1-4
Problem 2
Solving a Multi-Step Equation
Division Property of Equality
Got It? What is the solution of 3(2x 2 1) 2 2(3x 1 4) 5 11x?
Combine like terms.
Simplify.
12. The equation has been solved below. Use one of the reasons
in the box to justify each step.
Distributive Property
3(2x 2 1) 2 2(3x 1 4) 5 11x
6x 2 3 2 6x 2 8 5 11x
211 5 11x
11x
211
11 5 11
21 5 x
Problem 3
Using Properties of Equations to Solve Problems
Got It? The carpet at the right has perimeter 320 feet. What are the dimensions
x
of the carpet?
5 length of the carpet.
14. Use the formula for perimeter of a rectangle, P 5 2w 1 2/. Write an equation for
the perimeter of the carpet.
15. Solve for x.
16. The width of the carpet is
Problem 4
ft, and the length of the carpet is
ft.
Equations With No Solutions and Identities
Got It? Is the equation 7x 1 6 2 4x 5 12 1 3x 2 8 always, sometimes, or never true?
17. Combine like terms on each side of the equation.
1 6 5 3x 1
18. Now solve the equation.
19. The equation is always / sometimes / never true.
Chapter 1
16
3x
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13. Let x 5 the width of the carpet. Then
Problem 5 Solving a Literal Equation
Got It? The equation S 5 3F 2 24 relates shoe size S and length of a foot in inches F.
What is F in terms of S?
20. What two operations will you undo?
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21. Now solve the equation for F.
Lesson Check • Do you UNDERSTAND?
Reasoning Suppose you solve an equation and find that your school needs
4.3 buses for a class trip. Explain how to interpret this solution.
22. Why does the solution 4.3 not make sense as a solution to this problem?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
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23. What is the minimum number of buses that are needed for the trip? Explain.
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Math Success
Check off the vocabulary words that you understand.
equation
solution
inverse operations
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Lesson 1-4