Chapter 1.6, Problem 44PPSE

Solution

To solve the equation $4\left|3x+4\right|=4x+8$ .

  $x=-\frac{3}{2}\text{and }x=-1$

Given:

The equation, $4\left|3x+4\right|=4x+8$ .

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 $4\left|3x+4\right|=4x+8$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}4\left|3x+4\right|=4x+8\\ \frac{4\left|3x+4\right|}{4}=\frac{4x+8}{4}\\ \left|3x+4\right|=x+2\\ 3x+4=\pm \left(x+2\right)\\ 3x+4-4=\pm \left(x+2\right)-4\\ 3x=\pm \left(x+2\right)-4\\ 3x=-\left(x+2\right)-4\text{and }3x=\left(x+2\right)-4\\ 3x=-x-2-4\text{and }3x=x+2-4\\ 4x=-6\text{and }2x=-2\\ x=-\frac{6}{4}\text{and }x=-\frac{2}{2}\\ x=-\frac{3}{2}\text{and }x=-1\end{array}$

Thus, the solution of the equation are $x=-\frac{3}{2}\text{and }x=-1$ .