Problem 30 - Precalculus : Building Concepts and Connections

Question

Solve the rational inequality. $$\frac{4 x^{2}-9}{x+2}<0$$

Step-by-Step Solution

Answer

The solution of the inequality \(\frac{4 x^{2}-9}{x+2}<0\) is \((-∞,-2) \cup (-2, - \frac{3}{2}) \cup (\frac{3}{2},∞)\).

Step 1: Factor the numerator and denominator

The expression \(\frac{4 x^{2}-9}{x+2}\) can be factored into \(\frac{(2x-3)(2x+3)}{(x+2)}\) to simplify it.

Step 2: Identify the critical points

Critical points are values of \(x\) for which the numerator or denominator is zero. They divide the number line into intervals where the sign of the function might change. For this function, the critical points are \(x=-\frac{3}{2}\), \(x=\frac{3}{2}\) and \(x=-2\).

Step 3: Test intervals between critical points for the inequality sign

The critical points divide the values of \(x\) into four intervals: \((-∞,-2)\), \((-2, - \frac{3}{2})\), \( (-\frac{3}{2},\frac{3}{2})\), \((\frac{3}{2},∞)\). Pick a test point in each interval and substitute into the inequality. If the result is true, include this interval in the solution set. If the result is false, exclude this interval from the solution.

Step 4: Determine the solution

The intervals tested will give the solution set.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when solving rational inequalities. In this context, it involves breaking down a polynomial into simpler terms or factors that, when multiplied together, give the original polynomial. For instance, consider the expression \(4x^2 - 9\) found in the numerator of the rational inequality provided.

This can be recognized as a difference of squares, which follows the identity \(a^2 - b^2 = (a - b)(a + b)\).
Applying this, \(4x^2 - 9\) can be factored into \((2x-3)(2x+3)\), where \(a = 2x\) and \(b = 3\).

By factoring, we simplify the expression and make it easier to find critical points and solve the inequality.

• Factor each part of the inequality separately—both the numerator and the denominator.
• Look for common patterns such as difference of squares, perfect square trinomials, and the greatest common factor (GCF).
• Simplifying expressions by factoring is often the initial and most foundational step in solving polynomial equations and inequalities.

Critical Points

When dealing with rational inequalities, identifying critical points is essential. Critical points are values of \(x\) where the function can potentially change sign, and they occur where the numerator or the denominator equals zero.

In our example, the critical points of \(\frac{(2x-3)(2x+3)}{(x+2)}\) are determined by setting each factor equal to zero:

• For the numerator, set \((2x-3) = 0\) and \((2x+3) = 0\), resulting in \(x = \frac{3}{2}\) and \(x = -\frac{3}{2}\), respectively.
• For the denominator, set \(x + 2 = 0\), which gives \(x = -2\).

These critical points split the real number line into distinct intervals where the inequality might change its sign.

They are key for later steps because testing these intervals with values will help determine where the inequality holds true.
Remember, a critical point related to the denominator defines a point of discontinuity since division by zero is undefined.

Interval Testing

Once critical points are identified, interval testing comes into play. This strategy involves selecting test points in intervals between critical points to see where the inequality holds. Consider the intervals created by dividing the number line using identified critical points.

For our inequality, the critical points \((-\frac{3}{2}, \frac{3}{2}, -2)\) divide the number line into the intervals \((-∞,-2)\), \((-2, -\frac{3}{2})\), \((-\frac{3}{2},\frac{3}{2})\), \((\frac{3}{2},∞)\).

Here's how to proceed:

• Pick any test point within each interval. Ensure it's not a critical point.
• Substitute the test point into the inequality. For example, try \(x=-3\) in the interval \((-∞,-2)\).
• Check the sign of the resultant value. If the result satisfies the inequality \(< 0\), include that interval in the solution set. If not, exclude it.

Through testing, you reveal which parts of the number line satisfy the original inequality. This leads you to your final solution, showing where the inequality holds true across all possible values of \(x\).