Question

In Problems \(7-22,\) solve each inequality. 11\. \(x^{2}-9<0\)

Short answer

The solution set is \( ( -3, 3 ) \).

Step-by-step solution

Identify the inequality

The inequality given is \( x^{2} - 9 < 0 \). This can be rewritten to find the values of \( x \) that satisfy this inequality.

Rewrite the inequality

Rewrite the inequality in the form of a difference of squares: \( x^{2} - 9 = (x - 3)(x + 3) < 0 \).

Find critical points

Set each factor equal to zero and solve for \( x \): \( x - 3 = 0 \rightarrow x = 3 \) and \( x + 3 = 0 \rightarrow x = -3 \). These values divide the number line into intervals.

Test the intervals

Test values within each interval created by the critical points \( x = -3 \) and \( x = 3 \) to determine where the inequality \( (x - 3)(x + 3) < 0 \) holds true.

Choose test points

Choose points from the intervals: \[ (-\∞, -3) \]: test point \( x = -4 \); \[ (-3, 3) \]: test point \( x = 0 \); and \[ (3, \∞) \]: test point \( x = 4 \)

Analyze the test results

Evaluate \( (x - 3)(x + 3) \) at the test points: \( x = -4 \rightarrow (-4-3)(-4+3) = 7 \;( positive ) \), \( x = 0 \rightarrow (0-3)(0+3) = -9 \;( negative ) \), \( x = 4 \rightarrow (4-3)(4+3) = 7 \;( positive ) \). The inequality holds where the product is negative.

Determine the solution set

From the analysis, \( (x - 3)(x + 3) < 0\) is true in the interval \ ( -3, 3 ) \, excluding \ -3 \ and \ 3 \ because the inequality is strict ( < ).

Conclusion

Thus, the solution set for \( x^{2} - 9 < 0 \) is \ ( -3, 3 ) \.

Key concepts

Quadratic Inequality

A quadratic inequality involves a quadratic expression that is set to be less than, greater than, less than or equal to, or greater than or equal to a value. In this exercise, we have the inequality \(x^{2} - 9 < 0\). Quadratic inequalities generally look like \(ax^2 + bx + c < 0\) or similar forms.

To solve a quadratic inequality, follow these steps:

• Rewrite the inequality in a useful form.
• Find the critical points where the expression equals zero.
• Test the intervals between the critical points.

This process helps you determine which sections of the number line satisfy the inequality.

Critical Points

Critical points are the values of \(x\) that make the quadratic expression equal to zero. For the inequality \(x^{2} - 9 < 0\), the expression can be written as a product of factors:

\[ (x - 3)(x + 3) = 0 \]

By finding where each factor equals zero, we get the critical points:

• \(x - 3 = 0 \rightarrow x = 3\)
• \(x + 3 = 0 \rightarrow x = -3\)

These points, \(-3\) and \(3\), divide the number line into distinct intervals. They are essential for determining where the inequality changes its nature (i.e., from positive to negative or vice versa).

Interval Testing

Once we have determined the critical points, the next step is interval testing. This involves picking test points in each of the intervals divided by our critical points and checking if they satisfy the inequality.

For \(x^{2} - 9 < 0\), our critical points are \(-3\) and \(3\), dividing the number line into three intervals: \((-∞, -3)\), \((-3, 3)\), and \((3, ∞)\).

We choose representative test points like this:

• For \((-∞, -3)\): Test point is \(x = -4\)
• For \((-3, 3)\): Test point is \(x = 0\)
• For \((3, ∞)\): Test point is \(x = 4\)

We then evaluate the original inequality at these points to see if it holds.

Difference of Squares

The term 'difference of squares' refers to an expression that is the difference between two perfect squares. In our inequality, \(x^2 - 9\), we notice this can be written as:

\[ x^{2} - 9 = (x - 3)(x + 3) \]

This form is useful because it allows us to easily see the factors and critical points. Recognizing difference of squares can simplify solving inequalities by reducing complex expressions into easier-to-analyze factors. This factorization strategy transforms the quadratic inequality into a product of linear inequalities, making it easier to solve.