Chapter 1.6, Problem 49PPSE

Solution

To solve the equation $2\left|3x-7\right|=10x-8$ .

  $x=\frac{11}{8}$

Given:

The equation, $2\left|3x-7\right|=10x-8$ .

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 $2\left|3x-7\right|=10x-8$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}2\left|3x-7\right|=10x-8\\ \frac{2\left|3x-7\right|}{2}=\frac{10x-8}{2}\\ \left|3x-7\right|=5x-4\\ 3x-7=\pm \left(5x-4\right)\\ 3x=\pm \left(5x-4\right)+7\\ 3x=-\left(5x-4\right)+7\text{and }3x=\left(5x-4\right)+7\\ 3x=-5x+4+7\text{and }3x=5x-4+7\\ 8x=11\text{and }-2x=3\\ x=\frac{11}{8}\text{and }x=-\frac{3}{2}\end{array}$

Note that the right side of the given equation is negative for $x=-\frac{3}{2}$ , so it cannot be a solution of the equation.

Thus, the solution of the equation is $x=\frac{11}{8}$ .