Problem 12 - Algebra for College Students

Question

Solve each inequality and graph its solution set on a number line. $$x(x+3)(x-3) \leq 0$$

Step-by-Step Solution

Answer

The solution to the inequality is \([-3, 0] \cup [3, 3]\).

Step 1: Identify Critical Points

First, identify the critical points where the expression changes sign by setting each factor equal to zero.\[ x = 0, \quad x + 3 = 0 \implies x = -3, \quad x - 3 = 0 \implies x = 3 \]Thus, the critical points are \( x = 0, -3, \text{ and } 3 \).

Step 2: Determine Intervals

The critical points divide the number line into different intervals to test: \( (-\infty, -3) \), \( (-3, 0) \), \( (0, 3) \), and \( (3, \infty) \).

Step 3: Test Each Interval

Choose a test point from each interval and substitute it into the inequality to determine if the expression is positive or negative in that interval.- Interval \((-\infty, -3)\): Choose \(x = -4\). Then: \[ (-4)(-4+3)(-4-3) = (-4)(-1)(-7) = 28 > 0 \] (not in solution)- Interval \((-3, 0)\): Choose \(x = -1\). Then: \[ (-1)(-1+3)(-1-3) = (-1)(2)(-4) = 8 > 0 \] (not in solution)- Interval \((0, 3)\): Choose \(x = 1\). Then: \[ (1)(1+3)(1-3) = (1)(4)(-2) = -8 < 0 \] (included in solution)- Interval \((3, \infty)\): Choose \(x = 4\). Then: \[ (4)(4+3)(4-3) = (4)(7)(1) = 28 > 0 \] (not in solution)

Step 4: Test the Critical Points

Test the critical points \(x = -3, 0,\) and \(3\) in the inequality \(x(x+3)(x-3) \leq 0\) for inclusion.- \(x = -3:\) gives \((-3)(0)(-6) = 0\) (included)- \(x = 0:\) gives \((0)(3)(-3) = 0\) (included)- \(x = 3:\) gives \((3)(6)(0) = 0\) (included)

Step 5: Write the Solution

Combine the results for the intervals and critical points. The solution set is expressed in interval notation as:\[ [-3, 0] \cup [3, 3] \] or equivalently \([-3, 0] \cup \{3\}\).

Step 6: Graph the Solution

On a number line, shade the intervals from \(-3\) to \(0\) including both endpoints, and place a solid dot at \(x=3\). This represents the solution set graphically.

Key Concepts

Critical Points in Inequalities

Finding critical points in an inequality is like finding the key landmarks on a map. These points are where the expression changes its sign, which means it could go from positive to negative or vice versa. For example, in the inequality \[x(x+3)(x-3) \leq 0,\]the critical points are determined by setting each factor of the inequality to zero:

• \(x = 0\),
• \(x+3 = 0\ Rightarrow x = -3\),
• \(x-3 = 0\ Rightarrow x = 3\).

These points \(x = 0, -3,\) and \(3\) divide the number line and act as the transition points between intervals. At these points, the inequality expression equals zero. Critical points are essential because they help in determining the boundaries of the solution set.

Testing Intervals for Solutions

Once you've identified the critical points, the next step is to figure out what happens between these points. You've divided the number line into intervals based on these critical points. For our example, the critical points \(x = -3, 0,\) and \(3\) create the following intervals:

• \((-\infty, -3)\),
• \((-3, 0)\),
• \((0, 3)\),
• \((3, \infty)\).

Select a test point from each interval to substitute into the inequality. This helps in determining if each part is a solution or not:- In \((-\infty, -3)\), choose \(x = -4\). This results in a positive value, so this interval is not part of the solution.- For \((-3, 0)\), choosing \(x = -1\) also gives us a positive expression.- From \((0, 3)\), choose \(x = 1\), which makes the expression negative and thus part of the solution.- Lastly, in \((3, \infty)\), picking \(x = 4\) results in a positive outcome.By testing these intervals, you focus on where the inequality holds true, which is needed for correctly writing the solution set.

Graphing Solution Sets

Visualizing solutions is often the final step that solidifies your understanding of inequalities. After determining where the inequality is true by testing intervals and checking critical points, you can represent these solutions graphically on a number line.
For our example inequality \(x(x+3)(x-3) \leq 0\), the solution set found is \([-3, 0] \cup \{3\}\). This means the intervals from \(-3\) to \(0\), including both endpoints, satisfy the inequality, as does the individual point \(x = 3\).
On a number line:

• Shade the region from \(-3\) to \(0\) completely, including the endpoints, indicating these values satisfy the inequality.
• Place a solid dot at \(x = 3\) because the inequality allows for equality as well.

Through graphing, you provide a clear and concise visual of the solution set, making it much easier to understand which portions of the number line work for your inequality.