Chapter 4, Problem 2CR

To determine

The solution of the inequality $x^2\ge x$ and graph the solution set of the inequality.

Answer to Problem 2CR

Solution:

The solution of the inequality is $\left(-\infty ,0\right)\cup \left(1,\infty \right)$ . Graph of the solution set is

  

Explanation of Solution

Given information:

The inequality $x^2\ge x$

Rearrange the inequality, so that $0$ is on the right side

  $x^2-x\ge 0$ .

The above inequality is equivalent to the original inequality.

Now, graph the function $f\left(x\right)=x^2-x$ to find where $f\left(x\right)\ge 0$

Set $f\left(x\right)=x^2-x=0$

  $\Rightarrow x\left(x-1\right)=0$

  $\Rightarrow x=0,x-1=0$

  $\Rightarrow x=0,x=1$ .

The $x$ -intercept of the graph is $x=0,x=1$

Now, $f\left(0\right)=0^2-0=0$

Hence, $y=0$ is the $y$ -intercept of the graph

The function $f\left(x\right)$ is the parabola with vertex is at $x=-\frac{b}{2a}=-\frac{-1}{2\times 1}=1/2$ .

  $f\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^2-\frac{1}{2}=\frac{1}{4}-\frac{1}{2}=-\frac{1}{4}$ .

Thus, the vertex of the parabola is $\left(\frac{1}{2},-\frac{1}{4}\right)$

Hence, the graph of the function $f\left(x\right)=x^2-x$

  

Thus, the graph above the $x\text{-axis}$ is in the set $\left(-\infty ,0\right)$ and $\left(1,\infty \right)$

Hence, the graph of the solution set is