Problem 55 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$x^{3}-4 x>0$$

Step-by-Step Solution

Answer

The solution is \((-2, 0) \cup (2, \infty)\).

Step 1: Understand the Inequality

We are given the inequality \(x^3 - 4x > 0\). We want to find the range of values for \(x\) such that this inequality is true. This will involve determining when \(x^3 - 4x\) is positive.

Step 2: Find the Critical Points

To find where the expression changes sign (from positive to negative or vice versa), we set the equation \(x^3 - 4x = 0\). Factor the equation to solve it: \(x(x^2 - 4) = 0\). This gives us the critical points: \(x = 0\), \(x = 2\), and \(x = -2\).

Step 3: Determine the Intervals

The critical points divide the number line into intervals. The intervals to test will be \((-\infty, -2)\), \((-2, 0)\), \((0, 2)\), and \((2, \infty)\). We need to test a point from each interval to see if the expression is positive or negative in that interval.

Step 4: Test the Intervals

Choose a test point from each interval:- **Interval** \((-\infty, -2)\): Test point \(x = -3\). \[(-3)^3 - 4(-3) = -27 + 12 = -15 < 0.\] Expression is negative.- **Interval** \((-2, 0)\): Test point \(x = -1\). \[(-1)^3 - 4(-1) = -1 + 4 = 3 > 0.\] Expression is positive.- **Interval** \((0, 2)\): Test point \(x = 1\). \[(1)^3 - 4(1) = 1 - 4 = -3 < 0.\] Expression is negative.- **Interval** \((2, \infty)\): Test point \(x = 3\). \[(3)^3 - 4(3) = 27 - 12 = 15 > 0.\] Expression is positive.

Step 5: Write the Solution in Interval Notation

Based on our testing, the inequality \(x^3 - 4x > 0\) is satisfied in the intervals \((-2, 0)\) and \((2, \infty)\). Therefore, the solution in interval notation is \((-2, 0) \cup (2, \infty)\).

Step 6: Graph the Solution

To graph the solution set, draw a number line and shade the intervals \((-2, 0)\) and \((2, \infty)\). Mark open circles at \(x = -2\), \(x = 0\), and \(x = 2\) to indicate that these points are not included in the solution.

Key Concepts

Critical Points

In the context of nonlinear inequalities, critical points play a crucial role. They are points on the number line where the expression related to the inequality—here, the expression is \(x^3 - 4x\)—either becomes zero or changes its sign. To find these critical points, we first need to solve the related equation \(x^3 - 4x = 0\). The solution involves factoring: \(x(x^2 - 4) = 0\). This gives us the critical points \(x = 0\), \(x = 2\), and \(x = -2\). These points divide the number line into different regions or intervals. Each interval might have the expression positive or negative. Critical points are essential as they guide us to check where the inequality might hold true or become false.
Understanding critical points helps in uncovering how and where the sign of an inequality might change across the number line, which is essential in solving the inequality.

Interval Notation

Interval notation is a mathematical notation used to represent the solution set of an inequality. It provides a compact and clear way to state which numbers make the inequality true. After determining the critical points and testing intervals, we determine in which sections the inequality holds. In this exercise, the inequality \(x^3 - 4x > 0\) is satisfied in the intervals \((-2, 0)\) and \((2, \infty)\).
Interval notation uses brackets and parentheses to show the inclusivity or exclusivity of the endpoints. A parenthesis \((\) or \()\) means the endpoint is not included, which is the case here since the inequality is strict ('greater than' and not 'greater than or equal to'). This notation is efficient because it clearly shows not just the numbers but also the nature of their inclusion.

Testing Intervals

After identifying the critical points, the next step is testing intervals. This involves selecting a number from each interval between or beyond the critical points to determine where the inequality holds true. For example, if you have critical points at \(-2, 0, \) and \(2\), you create the intervals \((-\infty, -2)\), \((-2, 0)\), \((0, 2)\), and \((2, \infty)\).
In each of these intervals, pick any test point and substitute it into the inequality to check its truth. For example:

• In \((-\infty, -2)\), choose \(x = -3\); the expression is negative.
• In \((-2, 0)\), choose \(x = -1\); the expression is positive.
• In \((0, 2)\), choose \(x = 1\); the expression is negative.
• In \((2, \infty)\), choose \(x = 3\); the expression is positive.

Testing intervals helps ensure that the expression's sign is consistently positive, negative, or zero across the entire interval.

Sign Analysis

Sign analysis is a critical component when solving inequalities. It involves analyzing the sign (positive or negative) of the expression within each interval determined by critical points. To perform sign analysis, we substitute a test point from each interval into the inequality's expression to determine its sign.
From our testing earlier, we saw:

• The expression is negative in \((-\infty, -2)\) and \((0, 2)\).
• The expression is positive in \((-2, 0)\) and \((2, \infty)\).

The importance of sign analysis is that it helps us definitively determine the intervals where the original inequality holds true. By systematically checking each interval, we get a clear picture of where the solution lies, allowing us to accurately describe the solution in interval notation. This ensures that our solution captures all possible values for \(x\) that satisfy the inequality.