Chapter 1.6, Problem 80PPSE

Solution

Whether the absolute value inequality ${\left(\left|x\right|\right)}^2<x^2$ is always, sometimes, or never true.

Never True

Given:

The inequality ${\left(\left|x\right|\right)}^2<x^2$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• 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.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.
• The absolute value property: $\text{if }\left|x\right|<a,\text{then}-a<x<a$
• The absolute value property: $\text{if }\left|x\right|>a,\text{then}x>a\text{ or }x<-a$

Calculation:

Recall that the absolute value term always gives a positive output. So, the left side of the inequality ${\left(\left|x\right|\right)}^2<x^2$ must be positive.

Note that the right side of the inequality is always true as well, because the square of a real number is always positive. The value of the left side expression must be equal to the right-side expression.

Since a positive number cannot be less than itself, the given inequality is never true.

Thus, the given inequality is never true.