Chapter 5.2, Problem 13PPS

To determine

To solve the inequality:

  $\frac{1}{2}a<20$

And to graph the solution on a number line.

Answer to Problem 13PPS

  $\left(-\infty ,40\right)$

Explanation of Solution

Given:

  $\frac{1}{2}a<20$

Concept Used:

• 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.
• Properties of inequality:

1. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

To solve the inequality:

  $\frac{1}{2}a<20$

Multiply both sides of the inequality with 2, as

  $\begin{array}{c}\frac{1}{2}a<20\\ 2\times \frac{1}{2}a<2\times 20\\ a<40\end{array}$

Thus, the solution of given inequality is every number strictly less than 40, this can be represented in interval, as

  $\left(-\infty ,40\right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $40$ is strict, so use a hollow circle at 40. And the solution interval has $-\infty$ at left endpoint which means every number less than 40 to negative infinity is in the solution interval. The graph of the inequality is shown below: