Chapter 5.4, Problem 48SR

To determine

To solve the inequality:

  $5b-1\ge -11$

And to graph the solution on a number line.

Answer to Problem 48SR

  $\left[-2,\infty \right)$

  

Explanation of Solution

Given:

  $5b-1\ge -11$

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

  $5b-1\ge -11$

First, add 1 both sides of the inequality and simplify further using properties of the inequality, as

  $\begin{array}{c}5b-1\ge -11\\ \Rightarrow 5b-1+1>-11+1\\ \Rightarrow 5b\ge -10\\ \Rightarrow \frac{5b}{5}\ge -\frac{10}{5}\\ \Rightarrow b\ge -2\end{array}$

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

  $\left[-2,\infty \right)$

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 has $\infty$ at right endpoint which means every number greater than and equals to $-2$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: