Problem 39 - College Algebra with Modeling and Visualization

Question

Solve the inequality. $$ (x+4)(x-10) \leq 0 $$

Step-by-Step Solution

Answer

The solution is \([-4, 10]\).

Step 1: Find the roots of the inequality

To solve the inequality \((x+4)(x-10) \leq 0\), we first find the points where the expression equals zero. Set \( (x+4)(x-10) = 0 \). The solutions are \(x+4=0\) and \(x-10=0\), which gives the roots \(x=-4\) and \(x=10\).

Step 2: Determine the test intervals

The roots \(x=-4\) and \(x=10\) divide the number line into three intervals: \((-\infty, -4)\), \([-4, 10]\), and \((10, \infty)\). These intervals will be tested to determine where the inequality holds true.

Step 3: Test the intervals

Select a test point in each interval and substitute it into the inequality \((x+4)(x-10)\). 1. For \((-\infty, -4)\), choose \(x=-5\): \((x+4)(x-10) = (-5+4)(-5-10) = (-1)(-15) = 15\). Positive, so the inequality does not hold.2. For \([-4, 10]\), choose \(x=0\): \((0+4)(0-10) = (4)(-10) = -40\). Negative, so the inequality holds.3. For \((10, \infty)\), choose \(x=11\): \((11+4)(11-10) = (15)(1) = 15\). Positive, so the inequality does not hold.

Step 4: Include the roots in the solution

Since the inequality \((x+4)(x-10) \leq 0\) includes \( \leq \), the endpoints where the expression equals zero are included in the solution. Thus, both \(x=-4\) and \(x=10\) are solutions.

Key Concepts

Solving Quadratic Inequalities

Quadratic inequalities, unlike quadratic equations, do not merely seek specific root values. Instead, they establish conditions for an entire set of values that make the inequality true. In our example, the inequality \((x+4)(x-10) \leq 0\) looks for values of \(x\) which satisfy the expression being less than or equal to zero.
The first step in solving these inequalities is to determine the roots of the equation. These roots are the values where the inequality equals zero. Once identified, they help define regions on the number line where the inequality can be tested. By focusing on these intervals, we identify where inequalities turn true or false.
This approach effectively narrows down candidates for \(x\) and avoids a cumbersome graphing requirement, providing a streamlined pathway to understand where the inequality holds true along a continuous number line.

Roots of Equations

The roots of an equation are vital in dissecting inequalities because they denote the points where the quadratic expression shifts its sign. In quadratics, these roots are the solutions to the corresponding quadratic equation set to zero. For \((x+4)(x-10) = 0\), we find roots by solving \(x + 4 = 0\) and \(x - 10 = 0\).
The results are \(x = -4\) and \(x = 10\), which are not just numbers—they are the pivots that transition the inequality’s behavior. These pivot points allow us to subclassify the number line, defining intervals for further analysis to solve the inequality.
Understanding roots is crucial because they provide the boundaries where a quadratic expression transitions between positive, negative, or zero values, thus guiding the resolution of inequalities.

Number Line Intervals

Once the roots \(-4\) and \(10\) are established, they slice the number line into distinct intervals: \(( -\infty, -4)\), \([-4, 10]\), and \((10, \infty)\). These intervals are vital because they offer simplified, manageable segments to test which intervals satisfy the inequality.
We choose a test point from each interval to determine if the inequality holds. For instance, selecting \(x = 0\) in \([-4, 10]\), the expression \((0+4)(0-10) = -40\) is negative, fulfilling the inequality's requirement of being \( \leq 0\).
This method allows a logical breakdown of continuous number lines into evaluative sections, ensuring clarity while solving. Practicing this approach not only solidifies understanding but also builds confidence in tackling various inequalities.