Chapter 6, Problem 11QC

To determine

To solve the compound inequality:

  $a+1<4,\text{ or }a-1\ge 3$

And to graph the solution on a number line.

Answer to Problem 11QC

No solution.

Explanation of Solution

Given:

  $a+1<4,\text{ or }a-1\ge 3$

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:

  $a+1<4,\text{ or }a-1\ge 3$

First, solve the inequality

  $a+1<4$

By subtracting 1 from both sides, as

  $\begin{array}{c}a+1-1<4-1\\ \Rightarrow a<3\end{array}$

Now, solve the inequality

  $a-1\ge 3$

By adding 1 both sides, as

  $\begin{array}{c}a-1+1\ge 3+1\\ \Rightarrow a\ge 4\end{array}$

Combining both of solutions:

  $a<3,\text{ and }a\ge 4$

But there does not exist any number a which is less than 3 but greater than 4.

Thus, the given compound inequality has no solution.