Problem 11 - Algebra for College Students

Question

Solve each inequality and graph its solution set on a number line. $$x(x+2)(x-4) \leq 0$$

Step-by-Step Solution

Answer

The solution is \([-2, 4]\).

Step 1: Zero Product Property

To solve the inequality \(x(x+2)(x-4) \leq 0\), first identify where each factor equals zero. These are the points where the sign of the expression can change.\(x = 0\), \(x + 2 = 0\) gives \(x = -2\), and \(x - 4 = 0\) gives \(x = 4\).

Step 2: Critical Points and Number Line

Mark the critical points \(x = -2\), \(x = 0\), and \(x = 4\) on a number line. These points divide the number line into four intervals: \((-\infty, -2)\), \((-2, 0)\), \((0, 4)\), and \((4, \infty)\).

Step 3: Test Intervals

Choose test points from each interval to determine the sign of the expression in that interval. For \((-\infty, -2)\), use \(x = -3\); \((-3)(-3+2)(-3-4) = -21\), which is negative. For \((-2, 0)\), use \(x = -1\); \((-1)(-1+2)(-1-4) = 5\), which is positive. For \((0, 4)\), use \(x = 2\); \((2)(2+2)(2-4) = -16\), which is negative. For \((4, \infty)\), use \(x = 5\); \((5)(5+2)(5-4) = 35\), which is positive.

Step 4: Determine Valid Intervals

Since the inequality is \(x(x+2)(x-4) \leq 0\), we are looking for where the product is negative or zero. The valid intervals based on our tests are \([-2, 0]\) and \([0, 4]\), where the expression is either negative or zero.

Step 5: Graph the Solution

On the number line, shade the intervals \([-2, 0]\) and \([0, 4]\). Include closed circles at \(x = -2, 0,\) and \(4\) to show these points are included, as they make the expression zero, satisfying the inequality.

Key Concepts

Zero Product Property

The zero product property is a fundamental concept in algebra that helps solve equations of the form \(a \times b = 0\). Simply put, if the product of several terms equals zero, then at least one of the terms must be zero. This property is particularly useful for solving polynomial inequalities.
For the inequality \(x(x+2)(x-4) \leq 0\), the zero product property tells us to set each factor equal to zero. By doing so, we find the critical points where the expression becomes zero.

Finding Critical Points

When solving \(x(x+2)(x-4) \leq 0\), set each factor to zero.

• \(x = 0\)
• \(x + 2 = 0\) gives \(x = -2\)
• \(x - 4 = 0\) gives \(x = 4\)

These are the values where the expression can change its sign.

Critical Points

Critical points in the context of inequalities are the input values where the expression changes sign. These points divide the number line into different intervals, helping us understand where the expression is less than, greater than, or equal to zero.
In our inequality, the critical points are found using the zero product property. The values \(x = -2\), \(x = 0\), and \(x = 4\) divide the number line into intervals:

• \((-\infty, -2)\)
• \((-2, 0)\)
• \((0, 4)\)
• \((4, \infty)\)

These points are crucial as they indicate where the product changes from positive to negative or vice versa. They also indicate where the expression is exactly zero, which is critical when dealing with inequalities that include the equal sign.

Number Line Graphing

Graphing on a number line is a visual method for illustrating solutions to inequalities. This step involves plotting the critical points on a number line and marking the intervals according to the inequality.

Plotting Points and Intervals

For the inequality \(x(x+2)(x-4) \leq 0\), plot the critical points \(x = -2\), \(0\), and \(4\).

• These points separate the line into potential solution intervals: \([-2, 0]\) and \([0, 4]\).
• Shade the intervals where the expression meets the inequality condition (\(\leq 0\)), including endpoints since the inequality allows for zero.

This visual representation makes it easier to interpret where the inequality holds true. Closed circles are used at the endpoints to show they are included in the solution.

Interval Testing

Interval testing involves choosing test points within each interval formed by critical points. It's a systematic way to determine the sign of the expression within a specific range.
For our inequality, we select points in each interval:

• For \((-\infty, -2)\), choose \(x = -3\). The product is negative.
• For \((-2, 0)\), choose \(x = -1\). The product is positive.
• For \((0, 4)\), choose \(x = 2\). The product is negative.
• For \((4, \infty)\), choose \(x = 5\). The product is positive.

Determining Valid Intervals

Since the inequality looks for \(\leq 0\), focus on negative intervals: \([-2, 0]\) and \([0, 4]\). This means that in these intervals, the expression satisfies the inequality. These tests confirm and clarify the solution set, making sure all conditions are addressed correctly.