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Chapter 2: Problem 13

Solve the inequality. Then graph the solution set. $$x^{2}<9$$

Short Answer

Expert verified
The solution to the inequality \(x^{2}<9\) is \( -3 < x < 3\). Graphically, this is represented by a line on the number line between -3 and 3 (exclusive).

Step by step solution

01

Solve the quadratic inequality

To do this, first understand the inequality \(x^{2} < 9\). This could also be written as \(x^{2} - 9 < 0\). Now, think of the left side of the equation as a difference of squares. It factors to \((x - 3)(x + 3) < 0\).
02

Determine the roots

Now, set each factor equal to 0. Here, find that \(x - 3 = 0\) gives a root of \(x = 3\) and \(x + 3 = 0\) gives a root of \(x = -3\). These are the x-values where the curve of the quadratic function crosses the x-axis.
03

Test the intervals to find solutions

The values -3 and 3 split the number line into three intervals: \(-\infty to -3\), \(-3 to 3\), and \(3 to \infty\). Now, choose a test value from each interval, and substitute it into the factored inequality to determine if it's valid or not. Test values can be -4, 0, and 4, respectively. When testing, if the inequality is satisfied, this will mean that all values within that interval are solutions.
04

Graph the solution set

Plot the roots -3 and 3 on a number line. The solution set for this inequality will be the values between -3 and 3 (exclusive), because the quadratic function will be less than zero within that interval. This should be represented by an open circle at \(x = -3\) and \(x = 3\), and a line between these two points.

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