Chapter 1.6, Problem 24PPSE

Solution

To solve the equation $\left|2x+5\right|=3x+4$ and check the extraneous solutions.

  $x=1$

Given:

The equation, $\left|2x+5\right|=3x+4$ .

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 $\left|2x+5\right|=3x+4$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}\left|2x+5\right|=3x+4\\ 2x+5=\pm \left(3x+4\right)\\ 2x=\pm \left(3x+4\right)-5\\ 2x=-\left(3x+4\right)-5\text{and }2x=\left(3x+4\right)-5\\ 2x=-3x-4-5\text{and }2x=3x+4-5\\ 5x=-9\text{and }-x=-1\\ x=-\frac{9}{5}\text{and }x=1\end{array}$

Thus, the solution of the equation are $-\frac{9}{5}\text{and }1$ .

In order to check the solution, substitute back $x=-\frac{9}{5}\text{and }x=1$ into the given equation one by one and show that the left side expression is equal to the right-side expression as shown below

  $\begin{array}{c}\left|2x+5\right|=3x+4\\ \left|2\left(-\frac{9}{5}\right)+5\right|=3\left(-\frac{9}{5}\right)+4\\ \left|-\frac{18}{5}+5\right|=-\frac{27}{5}+4\\ \left|\frac{-18+25}{5}\right|=\frac{-27+20}{5}\\ \left|\frac{7}{5}\right|=\frac{-7}{5}\\ \frac{7}{5}=-\frac{7}{5}\text{False statement}\end{array}$

And,

  $\begin{array}{c}\left|2x+5\right|=3x+4\\ \left|2\left(1\right)+5\right|=3\left(1\right)+4\\ \left|2+5\right|=3+4\\ \left|7\right|=7\\ 7=7\text{True statement}\end{array}$

Thus, $x=1$ is the only solution of the equation.