Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

Solve the inequality. Graph the solution set.

$\left|2x-5\right|<3$

Step-by-Step Solution

Answer

The solution set of the given inequality is:

$\left(1,4\right)$

The graph of the solution set is given as:

Step 1. Given information

The given equation is $\left|2x-5\right|<3$

Step 2. Open the modulus function

We know that,  from the principle of absolute value,for $\left|x\right|<a$,  we have:

$-a<x<a$

Similarly, for $\left|2x-5\right|<3$, we have:

$-3<2x-5<3$

Step 3. Solving the inequality

Solving the inequality using properties, we get

$$\begin{gathered} -3<2x-5<3 \\ -3+5<2x<3+5 \\ 2<2x<8 \\ 1<x<4 \end{gathered}$$

Step 4. Verify the solution set

Substituting the value between $1$and $3$ in the inequality, we get:

$$\begin{gathered} \left|2x-5\right|<3 \\ \left|2\times 2-5\right|<3 \\ \left|4-5\right|<3 \\ \left|-1<3\right| \\ 1<3 \end{gathered}$$

This is true.

Hence, the solution set of the given inequality is $1<x<4$

Step 5. Graph the solution set

The graph of the solution set is given as: