Chapter 1.7, Problem 3E

(a)

To determine

To fill: The blank in the statement “The solution of the inequality $\left|x\right|\le 3$ is the interval _____”.

Answer to Problem 3E

The complete statement is “The solution of the inequality $\left|x\right|\le 3$ is the interval $\underset{\_}{\left[-3,3\right]}$”.

Explanation of Solution

Property used:

Properties of absolute value inequalities:

Inequality	Equivalent form
$\left|x\right|<c$	$-c<x<c$
$\left|x\right|\le c$	$-c\le x\le c$
$\left|x\right|>c$	$x<-c\text{or }c<x$
$\left|x\right|\ge c$	$x\le -c\text{or }c\le x$

Calculation:

The given inequality is $\left|x\right|\le 3$.

This is of the form $\left|x\right|\le c$.

Then by the properties mentioned above, the given inequality is equivalent to $-3\le x\le 3$.

Note that, here both the end points are included since the inequality sign is $\le$.

Then, the solution will be a closed interval. That is, $\left[-3,3\right]$.

Thus, the complete statement is “The solution of the inequality $\left|x\right|\le 3$ is the interval $\underset{\_}{\left[-3,3\right]}$”.

(b)

To determine

To fill: The blank in the statement “The solution of the inequality $\left|x\right|\ge 3$ is a union of two intervals _____ and ____”.

Answer to Problem 3E

The complete statement is “The solution of the inequality $\left|x\right|\ge 3$ is a union of two intervals $\underset{\_}{\left(-\infty ,-3\right]\cup \left[3,\infty \right)}$”.

Explanation of Solution

The given inequality is $\left|x\right|\ge 3$.

This is of the form $\left|x\right|\ge c$.

Then by the properties mentioned in part (a), the given inequality is equivalent to $x\le -3\text{or }3\le x$.

That is, the solution of the given inequality is the union of the set of all points that are less than or equal to $-3$, and the set of all points that are greater than or equal to 3.

Note that, here both the end points are included since the inequality sign is $\le$.

The set of all points that are less than or equal to $-3$, are denoted by the interval $\left(-\infty ,-3\right]$ and the set of all points that are greater than or equal to 3 is denoted by the interval $\left[3,\infty \right)$.

Thus, the complete statement is “The solution of the inequality $\left|x\right|\ge 3$ is a union of two interval, $\underset{\_}{\left(-\infty ,-3\right]\cup \left[3,\infty \right)}$”.