Exercise 51 Page 46

Before we can solve this equation, we need to isolate the absolute value expression using the Properties of Equality. An absolute value measures an expression's distance from a midpoint on a number line. |4x+7|= 32x+64This equation means that the distance is 32x+64, either in the positive direction or the negative direction. |4x+7|= 32x+64 ⇓ l4x+7= 32x+64 4x+7= - (32x+64) To find the solutions to the absolute value equation, we need to solve both of these cases for x.

|4x+7|=32x+64

lc 4x+7 ≥ 0:4x+7 = (32x+64) & (I) 4x+7 < 0:4x+7 = - (32x+64) & (II)

lc4x+7=32x+64 & (I) 4x+7= -(32x+64) & (II)

Distr

(II): Distribute - 1

l4x+7=32x+64 4x+7= -32x-64

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

l4x=32x+57 4x= -32x-71

SubEqn

(I): LHS-32x=RHS-32x

l- 28x=57 4x= -32x-71

AddEqn

(II): LHS+32x=RHS+32x

l-28x=57 36x= -71

DivEqn

(I): .LHS /(-28).=.RHS /(-28).

lx=- 5728 36x= -71

DivEqn

(II): .LHS /36.=.RHS /36.

lx = - 5728 x=- 7136

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 - 5728.

|4x+7|=32x+64

Substitute

x= -57/28

|4( -57/28)+7|? =32( -57/28)+64

▼

Simplify

MultPosNeg

a(- b)=- a * b

|- 4*57/28+7|? =- 32*57/28+64

MoveLeftFacToNum

a*b/c= a* b/c

|- 4* 57/28+7|? =- 32* 57/28+64

ReduceFrac

a/b=.a /4./.b /4.

|-57/7+7|? = -8* 57/7+64

Multiply

Multiply

|-57/7+7|? = -456/7+64

NumberToFrac

a = 7* a/7

|-57/7+49/7|? = -456/7+448/7

AddTerms

Add terms

|-8/7|? = -8/7

AbsNeg

|-8/7|=8/7

8/7 ≠ -8/7

We will check - 7136 in the same way.

|4x+7|=32x+64

Substitute

x= -71/36

|4( -71/36)+7|? =32( -71/36)+64

▼

Simplify

MultPosNeg

a(- b)=- a * b

|- 4*71/36+7|? =- 32*71/36+64

MoveLeftFacToNum

a*b/c= a* b/c

|- 4* 71/36+7|? =- 32* 71/36+64

ReduceFrac

a/b=.a /4./.b /4.

|- 71/9+7|? =- 8* 71/9+64

Multiply

Multiply

|-71/9+7|? = -568/9+64

NumberToFrac

a = 9* a/9

|-71/9+63/9|? = -568/9+576/9

AddFrac

Add fractions

|-8/9|? = 8/9

AbsNeg

|-8/9|=8/9

8/9 = 8/9

We see that - 7136 satisfies the original equation. However, - 5728 is an extraneous solution.