Chapter 5.3, Problem 20PPS

To determine

To solve the inequality:

  $3b-6\ge 15+24b$

And to graph the solution on a number line.

Answer to Problem 20PPS

  $\left(-\infty ,-1\right]$

  

Explanation of Solution

Given:

  $3b-6\ge 15+24b$

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

  $3b-6\ge 15+24b$

First, simplify this inequality, as

  $\begin{array}{c}3b-6\ge 15+24b\\ \Rightarrow 3b-6-3b\ge 15+24b-3b\\ \Rightarrow 0-6\ge 15+21b\\ \Rightarrow 0-6-15\ge 15+21b-15\\ \Rightarrow -21\ge 21b\end{array}$

Now to simplify further divide both side of the inequality by 2, as

  $\begin{array}{c}-21\ge 21b\\ \Rightarrow -\frac{21}{21}\ge \frac{21b}{21}\\ \Rightarrow -1\ge b\end{array}$

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

  $\left(-\infty ,-1\right]$

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 has $-\infty$ at right endpoint which means every number less than and equals to $-1$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: