Chapter 8.8, Problem 68CCR

To determine

Solve the inequality $3\le \frac{1}{2}+a$ and graph the solution on number line

Answer to Problem 68CCR

The solution of the inequality is $a\ge \frac{5}{2}$

The graph of the solution $a\ge \frac{5}{2}$ on number line is below:

  

Explanation of Solution

Given:

The inequality, $3\le \frac{1}{2}+a$

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
• Inequality sign get reverse on multiplying or dividing by a negative number

Calculation:

• In order to solve the mixed inequality $3\le \frac{1}{2}+a$ first isolate the variable term a by performing some basic operations. Here to isolate a terms on the left side, first subtract both sides by $\frac{1}{2}$ and then simplify further as shown below:

  $\begin{array}{c}3\le \frac{1}{2}+a\\ 3-\frac{1}{2}\le \frac{1}{2}+a-\frac{1}{2}\\ \frac{6-1}{2}\le a\\ \frac{5}{2}\le a\\ \Rightarrow a\ge \frac{5}{2}\end{array}$

Thus, the solution of the given inequality is $a\ge \frac{5}{2}$ .

The graph of the inequality solution $a\ge \frac{5}{2}$ is below: