Chapter 1.4, Problem 46E

To determine

To check: Whether the statement of the friend is correct or not for the absolute value equation $\left|3x+8\right|-9=-5$ explain with reasoning.

Explanation of Solution

Given:

The statement of friend is “In the absolute value equation $\left|3x+8\right|-9=-5$ has the constant on the right side of the equation is negative therefore it has no solution”.

Calculation:

The absolution value equation is $\left|3x+8\right|-9=-5$ .

Isolate the absolute value from the equation.

  $\begin{array}{c}\left|3x+8\right|-9+5=-5+5\\ \left|3x+8\right|-4=0\\ \left|3x+8\right|=4\end{array}$

The absolution function is represented below.

  $\left|x\right|=\{\begin{array}{l}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{array}$

Use the absolution function in the equation $\left|3x+8\right|=4$ .

  $\begin{array}{l}3x+8=4\text{ if }x\ge -\frac{8}{3}\\ -\left(3x+8\right)=4\text{ if }x<-\frac{8}{3}\end{array}$

Now solve for the first equation $3x+8=4$ .

  $\begin{array}{c}3x+8=4\\ 3x=4-8\\ 3x=-4\\ x=-\frac{4}{3}\end{array}$

Solve for the second equation $-\left(3x+8\right)=4$

  $\begin{array}{c}-\left(3x+8\right)=4\\ -3x-8=4\\ -3x=4+8\\ -3x=12\end{array}$

Further solve the above equation.

  $\begin{array}{c}-3x=12\\ x=-\frac{12}{3}\\ x=-4\end{array}$

Therefore, there are two solutions for $x$ that means the statement of the friend is incorrect.

Conclusion:

Thus, the statement of friend is incorrect.