Q. 5 - Precalculus Enhanced with Graphing Utilities

Question

Find the real solutions of the equation: 

$\left|2x+3\right|=4$

Step-by-Step Solution

Answer

The solution of the equation is :

$$\begin{gathered} x=\frac{1}{2} \\ x=\frac{-7}{2} \end{gathered}$$

Step 1. Given information

The given equation is $\left|2x+3\right|=4$

Step 2. Open the modulus function

We know that, for $\left|x\right|=a$, we have:

$$\begin{gathered} x=a \\ x=-a \end{gathered}$$

Similarly, for $\left|2x+3\right|=4$, we have:

$$\begin{gathered} 2x+3=4 \\ 2x+3=-4 \end{gathered}$$

Step 3. Solving the equations

Solving the equation $2x+3=4$, we get:

$$\begin{gathered} 2x+3=4 \\ 2x=4-3 \\ 2x=-1 \\ x=-\frac{1}{2} \end{gathered}$$

Solving the equation $2x+3=-4$, we get:

$$\begin{gathered} 2x+3=-4 \\ 2x=-4-3 \\ 2x=-7 \\ x=-\frac{7}{2} \end{gathered}$$

Step 4. Verify the solutions

Substituting $x=\frac{1}{2}$ in the given equation,  we get:

$$\begin{gathered} \left|2x+3\right|=4 \\ \left|2\times \frac{1}{2}+3\right|=4 \\ \left|1+3\right|=4 \\ \left|4\right|=4 \\ 4=4 \end{gathered}$$

This is true.

Substituting $x=-\frac{7}{2}$ in the given equation, we get:

$$\begin{gathered} \left|2x+3\right|=4 \\ \left|2\times \left(-\frac{7}{2}\right)+3\right|=4 \\ \left|-7+3\right|=4 \\ \left|-4\right|=4 \\ 4=4 \end{gathered}$$

This is true.

Thus, the solution of the equation is 

$$\begin{gathered} x=\frac{1}{2} \\ x=-\frac{7}{2} \end{gathered}$$