Question

solve |5x + 60| = 10x. identify the solution and an extraneous solution.
a. solution: x = -4;
extraneous solution: x = 12

b. solution: x = 12;
extraneous solution: x = -12

c. solution: x = 4;
extraneous solution: x = -12

d. solution: x = 12;
extraneous solution: x = -4

Explanation:

Step1: Split absolute value equation

Case1: $5x + 60 = 10x$
Case2: $5x + 60 = -10x$

Step2: Solve Case1

$$\begin{align} 5x + 60 &= 10x \\ 60 &= 10x - 5x \\ 60 &= 5x \\ x &= \frac{60}{5} = 12 \end{align}$$

Step3: Solve Case2

$$\begin{align} 5x + 60 &= -10x \\ 5x + 10x &= -60 \\ 15x &= -60 \\ x &= \frac{-60}{15} = -4 \end{align}$$

Step4: Verify solutions

For $x=12$: $|5(12)+60|=|60+60|=120$, $10(12)=120$. Valid.
For $x=-4$: $|5(-4)+60|=|-20+60|=40$, $10(-4)=-40$. $40
eq-40$. Extraneous.

Answer:

D. Solution: $x = 12$; extraneous solution: $x = -4$