Chapter 1.5, Problem 4LC

Solution

To solve the inequality $3<5x-2<7$ and graph the solution.

  $1<x<\frac{9}{5}$

The graph of solution set is shown below:

  

Given:

The inequality, $3<5x-2<7$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$ , use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$ , use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to solve the given inequality $3<5x-2<7$ , perform the basic algebraic operations to isolate the variable term as shown below

  $\begin{array}{c}3<5x-2<7\\ 3+2<5x-2+2<7+2\\ 5<5x<9\\ \frac{5}{5}<\frac{5x}{5}<\frac{9}{5}\\ 1<x<\frac{9}{5}\end{array}$

Thus, the solution of the given inequality is $1<x<\frac{9}{5}$ .

Since the inequality sign is strict at both endpoints, use a hollow circle at both endpoints and shade the region between them as shown below