Chapter 5.4, Problem 4CYU

To determine

To solve below compound inequality:

  $2\le g+4<7$

And to graph the solution on a number line.

Answer to Problem 4CYU

  $\left[-2,3\right)$

  

Explanation of Solution

Given:

  $2\le g+4<7$

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:

  $2\le g+4<7$

Subtract 4 from each side of the inequality, as

  $\begin{array}{c}2\le g+4<7\\ \Rightarrow 2-4\le g+4-4<7-4\\ \Rightarrow -2\le g+0<3\\ \Rightarrow -2\le g<3\end{array}$

Thus, the solution of given inequality is every number greater than and equals to $-2$ but strictly less than 3, this can be represented in interval form, as

  $\left[-2,3\right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, $-2$ , is not strict, so use a solid circle at $-2$ . And the solution interval at right endpoint, 3, is strict so use a hollow circle at 3. Which means every number greater than and equals to $-2$ but strictly less than 3 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: