Question

Solve each equation. Then graph the solution set.

$\left|-4w+2\right|=14$

Short answer

The Solution set is $\{-3,4\}$.

Step-by-step solution

Step 1. Case when the modulus expression is positive.

The absolute value or modulus of a real number x, denoted $\left|x\right|$, it is the non-negative value of $x$without regard to its sign.

$\left|x\right|=x$if$x$ is positive,

$\left|x\right|=-x$if$x$ is negative,

$\left|0\right|=0$.

For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5.

We are given the expression$|-4w+2|=14$ assuming a positive value for the modulus expression we get:

role="math" localid="1643046058341" $\begin{array}{l}-4w+2=14\\ -4w+2-2=14-2\left[\text{subtract\hspace{0.33em}}2\text{\hspace{0.33em}from}bothsides\right]\\ -4w=12\\ w=\frac{12}{-4}\\ w=-3\end{array}$

Thus$w=-3$ is one of the solutions.

Step 2. Case when the modulus expression is negative.

$\begin{array}{l}-4w+2=-14\\ -4w+2-2=-14-2\left[\text{subtract\hspace{0.33em}}2\text{\hspace{0.33em}from}bothsides\right]\\ -4w=-16\left[\text{divide\hspace{0.33em}by-}4\text{\hspace{0.33em}from}bothsides\right]\\ \frac{-4w}{-4}=\frac{-16}{-4}\\ w=4\end{array}$

Thus$w=4$ is one of the solutions.

Step 3. Graphing the solution set.

The solution set is $\{-3,4\}$, we graph these values on the number line as shown below: