Chapter 4.6, Problem 30AYU

In Problems 19-48, solve each inequality algebraically.

$x^4\text{ < 9}{x}^2$

To determine

To find: The inequality algebraically.

Answer to Problem 30AYU

$\left\{x|-3<x<0,0<x<3\right\}$

Explanation of Solution

Given:

$x^4<9x^2$

$x^4-9x^2<0$

The zero of $f\left(x\right)=x^4-9x^2=x^2\left(x^2-9\right)=x^2\left(x+3\right)\left(x-3\right)$ is $x=-3,0,3$.

Use the zeros to separate the real number line into intervals.

$\left(-\infty ,-3\right),\left(-3,0\right),\left(0,3\right),\left(3,\infty \right)$

Select a test number in each interval found in above and evaluate $f\left(x\right)=x^2\left(x+3\right)\left(x-3\right)$ each number to determine if $f\left(x\right)$ is positive or negative.

Interval	$\left(-\infty ,-3\right)$	$\left(-3,0\right)$	$\left(0,3\right)$	$\left(3,\infty \right)$
Number chosen	$-4$	$-2$	1	4
Value of $f$	112	$-20$	$-8$	112
Conclusion	Positive	Negative	Negative	Positive

Since we want to know where $f\left(x\right)$ is negative, conclude that $f\left(x\right)<0$. The solution set is $\left\{x|-3<x<0,0<x<3\right\}$ or, using interval notation $\left(-3,0\right)\cup \left(0,3\right)$.