Question

Solve each equation. Then graph the solution set.

$\left|\frac{3}{4}a-3\right|=9$

Short answer

The equation $\left|\frac{3}{4}a-3\right|=9$has solution set as $a=16$and $a=-8$and the graph of the solution set is:

Step-by-step solution

Step-1. Define absolute value equation.

For any real numbers a and b, if$\left|a\right|=b$ and $b\ge 0$, then$a=b$ or $a=-b$.

Step-2. Formulate case 1 when 34a−3=9.

Simplify the expression for a.

$$\begin{gathered} \frac{3}{4}a-3+3=9+3 \\ \frac{3}{4}a=12 \\ \frac{3}{4}a\times \frac{4}{3}=12\times \frac{4}{3} \\ a=16 \end{gathered}$$

Step-3. Formulate case 2 when 34a−3=−9.

Simplify the expression for a.

$$\begin{gathered} \frac{3}{4}a-3+3=-9+3 \\ \frac{3}{4}a=-6 \\ \frac{3}{4}a\times \frac{4}{3}=-6\times \frac{4}{3} \\ a=-8 \end{gathered}$$

Step-4. Plot the graph of the solution set.

From case 1 and case 2, $a=-8,16$. Plot these values on the number line.