I had problems with question 8 too. Unit1 didn’t show us the proper way for solving absolute value equations. You will find the correct answer by constructing piecewise definitions and evaluating the ranges. I will show this technique here.

-5|2+4X| = -32(X+3/4) - |X| +1

*Step 1:*

Take the expression that is inside the absolute value bars and set that expression equal to zero. Then solve for X. (This value for X is the critical value, because X changes sign at this value.)

For |2+4X| :

2 + 4X = 0

X = - 1/2

For |X| :

X = 0

*Step 2:*

Determine the sign of the expression inside the absolute value bars on both sides of the critical value. This is done by testing a value for X on each side of that critical value.

For 2 + 4X :

X = -1 --> 2 - 4 = -2 --> So |2 + 4X| is negative if X < -1/2

X = 1 --> 2 + 4 = 6 --> So |2 + 4X| is positive if X ≥ - 1/2 (The ≥ sign includes 0 as a positive number)

For X :

X = -1 --> So |X| is negative if X < 0

X = 1 --> So |X| is positive if X ≥ 0 (Again, the ≥ sign includes 0 as a positive number)

*Step 3:*

Construct piecewise definitions.

|2 + 4X|=

-(2 + 4X) if X < - 1/2 {A}

(2 + 4X) if X ≥ - 1/2 {B}

|X| =

-(X) if X < 0 {C}

(X) if X ≥ 0 {D}

*Step 4:*

Determine the ranges and evaluate the expression for the ranges.

There are 3 ranges:

X < - 1/2

Evaluate for X < - 1/2 (use {A} and {C}) : -5 (-2-4X) = -32 (X + 3/4) - (-X) + 1

This will result in X = - 11/17

-11/17 < -1/2, so the solution is correct!

Evaluate for -1/2 ≤ X < 0 (use {B} and {C}) : -5 (2+4X) = -32 (X + 3/4) - (-X) + 1

This will result in X = - 13/11

- 13/11 < - 1/2, so the solution is false!

Evaluate for X > 0 (use {B} and {D}) : -5 (2+4X) = -32 (X + 3/4) - X + 1

This will result in X = -1

-1 < 0, so the solution is false!

**The correct answer is - 11/17 !!!**