Chapter 5.4, Problem 11PPS

To determine

To solve below compound inequality:

  $-3\le 7c+4<18$

And to graph the solution on a number line.

Answer to Problem 11PPS

  $\left[-1,2\right)$

  

Explanation of Solution

Given:

  $-3\le 7c+4<18$

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 compound inequality:

  $-3\le 7c+4<18$

Subtract 4 from each side of the inequality, as

  $\begin{array}{c}-3\le 7c+4<18\\ \Rightarrow -3-4\le 7c+4-4<18-4\\ \Rightarrow -7\le 7c+0<14\\ \Rightarrow -7\le 7c<14\end{array}$

Now, to simplify further divide each side by 7, as

  $\begin{array}{c}-7\le 7c<14\\ \Rightarrow -\frac{7}{7}\le \frac{7c}{7}<\frac{14}{7}\\ \Rightarrow -1\le c<2\end{array}$

Thus, the solution of given inequality is every number greater than and equals to $-1$ but strictly less than 2, this can be represented in interval form, as

  $\left[-1,2\right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, $-1$ , is not strict, so use a solid circle at $-1$ . And the solution interval at right endpoint, 2, is strict so use a hollow circle at 2. The graph of the solution of inequality is shown below: