Chapter 1.6, Problem 3LC

Solution

To solve the equation $\left|x-2\right|=4x+8$ and check the answer.

  $x=-\frac{6}{5}$

Given:

The equation, $\left|x-2\right|=4x+8$ .

Concept Used:

• The opposite operation of addition is subtraction and the opposite operation of subtraction is addition.
• The opposite operation of multiplication is division and the opposite operation of division is multiplication.
• Distributive property: $a\left(b+c\right)=a\cdot b+a\cdot c$
• The absolute value property: $\text{if }\left|x\right|=a,\text{then}x=\pm a$

Calculation:

In order to solve the given equation $\left|x-2\right|=4x+8$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}\left|x-2\right|=4x+8\\ x-2=\pm \left(4x+8\right)\\ x-2+2=\pm \left(4x+8\right)+2\\ x=\pm \left(4x+8\right)+2\\ x=-\left(4x+8\right)+2\text{and }x=\left(4x+8\right)+2\\ x=-4x-8+2\text{and }x=4x+10\\ 5x=-6\text{and }-3x=10\\ x=-\frac{6}{5}\text{and }x=-\frac{10}{3}\end{array}$

Thus, the solution of the equation are $-\frac{6}{5}\text{and }-\frac{10}{3}$ .

In order to check the solution, substitute back $x=-\frac{6}{5}\text{and }x=-\frac{10}{3}$ into the given equation one by one and show that the left side expression is equal to the right-side expression as shown below

  $\begin{array}{c}\left|x-2\right|=4x+8\\ \left|-\frac{6}{5}-2\right|=4\left(-\frac{6}{5}\right)+8\\ \left|\frac{-6-10}{5}\right|=-\frac{24}{5}+8\\ \left|\frac{-16}{5}\right|=\frac{-24+40}{5}\\ \frac{16}{5}=\frac{16}{5}\text{True statement}\end{array}$

And

  $\begin{array}{c}\left|x-2\right|=4x+8\\ \left|-\frac{10}{3}-2\right|=4\left(-\frac{10}{3}\right)+8\\ \left|\frac{-10-6}{3}\right|=-\frac{40}{3}+8\\ \left|\frac{-16}{3}\right|=\frac{-40+24}{3}\\ \frac{16}{3}=-\frac{16}{3}\text{False statement}\end{array}$

Thus, $x=-\frac{6}{5}$ is the solution of the equation.