Exercise 49 Page 46

Before we can solve this equation, we need to isolate the absolute value expression using the Properties of Equality.

2|3x-7|=10x-8

DivEqn

.LHS /2.=.RHS /2.

|3x-7|=10x-8/2

WriteDiffFrac

Write as a difference of fractions

|3x-7|=10x/2-8/2

CalcQuot

Calculate quotient

|3x-7|=5x-4

An absolute value measures an expression's distance from a midpoint on a number line. |3x-7|= 5x-4This equation means that the distance is 5x-4, either in the positive direction or the negative direction. |3x-7|= 5x-4 ⇒ l3x-7= 5x-4 3x-7= - (5x-4) To find the solutions to the absolute value equation, we need to solve both of these cases for x.

|3x-7|=5x-4

lc 3x-7 ≥ 0:3x-7 = (5x-4) & (I) 3x-7 < 0:3x-7 = - (5x-4) & (II)

lc3x-7=5x-4 & (I) 3x-7= -(5x-4) & (II)

Distr

(II): Distribute - 1

l3x-7=5x-4 3x-7= -5x+4

(I), (II): LHS+7=RHS+7

l3x=5x+3 3x= -5x+11

SubEqn

(I): LHS-5x=RHS-5x

l-2x=3 3x= -5x+11

AddEqn

(II): LHS+5x=RHS+5x

l-2x=3 8x=11

DivEqn

(I): .LHS /(-2).=.RHS /(-2).

lx=- 32 8x=11

DivEqn

(II): .LHS /8.=.RHS /8.

lx=- 32 x = 118

Checking Our Answers

When solving an absolute value equation, it is important to check for extraneous solutions. We can check our answers by substituting them back into the original equation. Let's start with - 32.

|3x-7|=5x-4

Substitute

x= -3/2

|3( -3/2)-7|? =5( -3/2)-4

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Simplify

MultPosNeg

a(- b)=- a * b

|- 3*3/2-7|? =- 5*3/2-4

MoveLeftFacToNum

a*b/c= a* b/c

|-9/2-7|? =-15/2-4