Chapter P.7, Problem 44E

Solution

The solution set of the absolute value inequality $x^2>x$ .

  $\text{A}.\left(-\infty ,0\right)\cup \left(1,\infty \right)$

Given:

The absolute value inequality $x^2>x$ .

Concept used:

• Absolute value inequality states that, if $\left|x\right|<a$ , then the solutions are $-a<x<a$ .
• The value of an absolute value function is always greater than or equal to 0.
• 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.

Calculation:

In order to solve the given absolute value inequality, use the definition of absolute value function and simplify further as shown below

  $\begin{array}{c}{x}^2>x\\ x^2-x>x-x\\ x^2-x>0\\ x\left(x-1\right)>0\\ \Rightarrow x<0\text{ or }x>1\end{array}$

Thus, the solution set of the given absolute value inequality is $\left(-\infty ,0\right)\cup \left(1,\infty \right)$ .