Chapter 2.3, Problem 23E

To determine

To solve the inequality:

$2\le -\frac{2}{9}x$

And to graph the solution on a number line.

Answer to Problem 23E

  $\left(-\infty ,-9\right]$

Explanation of Solution

Given:

$2\le -\frac{2}{9}x$

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.$b\le c\Rightarrow b\pm a\le c\pm a$

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

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

Calculation:

In order to solve the inequality

$2\le -\frac{2}{9}x$

Multiply both sides of the inequality by 9, as

  $\begin{array}{c}2\times 9\le 9\times \left(-\frac{2}{9}x\right)\\ \Rightarrow 18\le -2x\end{array}$

And then divide both sides by that would change the inequality, as

  $\begin{array}{c}\frac{18}{-2}\ge -\frac{2x}{-2}\\ \Rightarrow -9\ge x\end{array}$

Thus, the solution of given inequality is every number strictly less than and equals to $-9$ , this can be represented in interval form, as

  $\left(-\infty ,-9\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $-9$ is not strict, so use a solid circle at $-9$ . And the solution interval has $-\infty$ at right endpoint which means every number less than and equals to $-9$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: