Chapter 1.5, Problem 46PPSE

Solution

To solve the inequality $6\left(x-2.5\right)\ge 8-6\left(3.5+x\right)$ and graph the solution.

  $x\ge \frac{1}{6}$

The graph of solution set is shown below:

  

Given:

The compound inequality, $6\left(x-2.5\right)\ge 8-6\left(3.5+x\right)$ .

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 $6\left(x-2.5\right)\ge 8-6\left(3.5+x\right)$ , perform the basic algebraic operations to isolate the variable term as shown below

  $\begin{array}{c}6\left(x-2.5\right)\ge 8-6\left(3.5+x\right)\\ 6x-15\ge 8-21-6x\\ 6x-15\ge -13-6x\\ 6x+6x-15\ge -13-6x+6x\\ 12x-15\ge -13\\ 12x-15+15\ge -13+15\\ 12x\ge 2\\ \frac{12x}{12}\ge \frac{2}{12}\\ x\ge \frac{1}{6}\end{array}$

Thus, the solution of the given inequality is $x\ge \frac{1}{6}$ .

Since the inequality sign is not strict, use the solid circles at $\frac{1}{6}$ and shade the region to the right side as shown below