Chapter 8 - XYZ Custom Plus

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488<strong>Chapter</strong> 8 Linear Equations and InequalitiesSimplify each expression.4. 2x + 6 + 3x − 5Example 4Simplify the expression 3x + 5 + 2x − 3.Solution 3x + 5 + 2x − 3 = 3x + 2x + 5 − 3 Commutative property= (3x + 2x) + (5 − 3) Associative property= (3 + 2)x + (5 − 3) Distributive property= 5x + 2 Addition5. 8a − 2 − 4a + 5Example 5Simplify the expression 4a − 7 − 2a + 3.Solution 4a − 7 − 2a + 3 = (4a − 2a) + (−7 + 3) Commutative and associativeproperties= (4 − 2)a + (−7 + 3) Distributive property= 2a − 4 Addition6. 9x + 1 − x − 6Example 6Simplify the expression 5x + 8 − x − 6.Solution5x + 8 − x − 6 = (5x − x) + (8 − 6) Commutative and associative properties= (5 − 1)x + (8 − 6) Distributive property= 4x + 2 AdditionBSimplifying Expressions Containing ParenthesesIf an expression contains parentheses, it is often necessary to apply the distributiveproperty to remove the parentheses before combining similar terms.7. Simplify the expression.4(3x − 5) − 2Example 7Simplify the expression 5(2x − 8) − 3.Solution We begin by distributing the 5 across 2x − 8. We then combine similarterms:5(2x − 8) − 3 = 10x − 40 − 3 Distributive property= 10x − 438. Simplify 5 − 2(4y + 1).Example 8Simplify 7 − 3(2y + 1).Solution By the rule for order of operations, we must multiply before weadd or subtract. For that reason, it would be incorrect to subtract 3 from 7 first.Instead, we multiply −3 and 2y + 1 to remove the parentheses and then combinesimilar terms:7 − 3(2y + 1) = 7 − 6y − 3 Distributive property= −6y + 49. Simplify 6(x − 3) − (7x + 2).Example 9Simplify 5(x − 2) − (3x + 4).Solution We begin by applying the distributive property to remove the parentheses.The expression −(3x + 4) can be thought of as −1(3x + 4). Thinking of itin this way allows us to apply the distributive property:−1(3x + 4) = −1(3x) + (−1)(4)= −3x − 4Answers4. 5x + 1 5. 4a + 3 6. 8x − 57. 12x − 22 8. −8y + 39. −x − 20The complete solution looks like this:5(x − 2) − (3x + 4) = 5x − 10 − 3x − 4 Distributive property= 2x − 14 Combine similar terms
8.1 Simplifying Expressions489As you can see from the explanation in Example 9, we use the distributiveproperty to simplify expressions in which parentheses are preceded by a negativesign. In general we can write−(a + b) = −1(a + b)= −a + (−b)= −a − bThe negative sign outside the parentheses ends up changing the sign of eachterm within the parentheses. In words, we say “the opposite of a sum is the sumof the opposites.”C The Value of an ExpressionRecall from <strong>Chapter</strong> 1, an expression like 3x + 2 has a certain value depending onwhat number we assign to x. For instance, when x is 4, 3x + 2 becomes 3(4) + 2,or 14. When x is −8, 3x + 2 becomes 3(−8) + 2, or −22. The value of an expressionis found by replacing the variable with a given number.ExamplesFind the value of the following expressions by replacingthe variable with the given number.Value ofValue ofExpression the Variable the Expression10. 3x − 1 x = 2 3(2) − 1 = 6 − 1 = 511. 7a + 4 a = −3 7(−3) + 4 = −21 + 4 = −1712. 2x − 3 + 4x x = −1 2(−1) − 3 + 4(−1)= −2 − 3 + (−4) = −913. 2x − 5 − 8x x = 5 2(5) − 5 − 8(5)= 10 − 5 − 40 = −3514. y 2 − 6y + 9 y = 4 4 2 − 6(4) + 9 = 16 − 24 + 9 = 110. Find the value of 4x − 7 whenx = 3.11. Find the value of 2a + 4 whena = −5.12. Find the value of 2x − 5 + 6xwhen x = −2.13. Find the value of 7x − 3 − 4xwhen x = 10.14. Find the value of y 2 − 10y + 25when y = −2.Simplifying an expression should not change its value; that is, if an expressionhas a certain value when x is 5, then it will always have that value no matter howmuch it has been simplified, as long as x is 5. If we were to simplify the expressionin Example 13 first, it would look like2x − 5 − 8x = −6x − 5When x is 5, the simplified expression −6x − 5 is−6(5) − 5 = −30 − 5 = −35It has the same value as the original expression when x is 5.We also can find the value of an expression that contains two variables if weknow the values for both variables.Answers10. 5 11. −6 12. −21 13. 2714. 49
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RESEARCH PROJECTStand and DeliverTh
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