Chapter 4.3, Problem 1QR

To determine

To calculate: The solution of the inequality $x^2-9<0$ .

Answer to Problem 1QR

The solution of the inequality $x^2-9<0$ lies in the interval where $x\in \left(-3,3\right)$ .

Explanation of Solution

Given information: The quadratic inequality $x^2-9<0$ .

Calculation:

The corresponding quadratic equation is $x^2-9=0$ .

  $x^2-9=0$   ...... (1)

Calculate the possible values of $x$ .

  $\begin{array}{c}{x}^2-9=0\\ \left(x-3\right)\left(x+3\right)=0\end{array}$

Set each value to $0$ .

  $\begin{array}{c}x-3=0\\ x=3\end{array}$

And,

  $\begin{array}{c}x+3=0\\ x=-3\end{array}$

Now check for the points where the expression is less than $0$ , that is to find the value where expression is negative.

Create a table of signs.

Interval	Value of $x$	Value of $x^2-9$	sign
$\left(-\infty ,-3\right)$	$-4$	$\begin{array}{c}{\left(-4\right)}^2-9=16-9\\ =7\\ >0\end{array}$	Positive
$\left(-3,3\right)$	0	$\begin{array}{c}{\left(0\right)}^2-9=0-9\\ =-9\\ <0\end{array}$	Negative
$\left(3,\infty \right)$	$4$	$\begin{array}{c}{\left(4\right)}^2-9=16-9\\ =7\\ >0\end{array}$	Positive

Here,

• $x^2-9$ is negative for the interval $\left(-3,3\right)$ .

The value of inequality $x^2-9<0$ is true for $x\in \left(-3,3\right)$ .

Thus, the solution of the inequality $x^2-9<0$ lies in the interval where $x\in \left(-3,3\right)$ .