Question

Solve the equation. Check the solutions.

$3|2a+7|=3a+12$

Short answer

The values are $a=-3,-\frac{11}{3}$.

Step-by-step solution

Step 1- Apply the concept of absolute value.

Absolute value describes the distance from zero that a number is on the number line,without considering direction. The absolute value of a number is never negative.

Step 2- Simplify the expression.

Consider the expression $3|2a+7|=3a+12$.

Divide both sides of the equation by 3 as follows:

$\begin{array}{l}3|2a+7|=3a+12\\ |2a+7|=a+4\end{array}$

Step 3- Step description.

Apply the absolute rule as follows:

Case 1. When $2a+7=-\left(a+4\right)$.

$\begin{array}{l}2a+7=-(a+4)\\ 2a+7=-a-4\\ 2a+a=-4-7\\ 3a=-11\\ a=-\frac{11}{3}\end{array}$

Case 2. When $2a+7=\left(a+4\right)$

$\begin{array}{l}2a+7=(a+4)\\ 2a-a=4-7\\ a=-3\end{array}$

Therefore, the value is $a=-3,-\frac{11}{3}$.

Step 4-Verify the solutions.

Substitute $a=-3,-\frac{11}{3}$in the equation $3|2a+7|=3a+12$and simplify as follows:

$\begin{array}{l}3|2a+7|=3a+12\\ 3|2(-3)+7|=3(-3)+12\\ 3|-6+7|=-9+12\\ 3\left|1\right|=3\\ 3=3\end{array}$

$\begin{array}{l}3|2a+7|=3a+12\\ 3|2(-\frac{11}{3})+7|=3(-\frac{11}{3})+12\\ 3|-\frac{22}{3}+7|=-11+12\\ 3\left|\frac{-22+21}{3}\right|=1\\ 3|-\frac{1}{3}|=1\\ 1=1\end{array}$

Since the left-hand side and right-hand side is equal in both the cases thus the values are $a=-3,-\frac{11}{3}$.