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Solve the following inequality.

$(x-8)(x-2)(x+4) \geq 0$



Write your answer as an interval or union of intervals.

Verified Answer :

  • Identify the roots of the cubic polynomial: \(x = -4, 2, 8\).
  • Divide the number line into the intervals: \((-\infty, -4]\), \([-4, 2]\), \([2, 8]\), and \([8, \infty)\).
  • Test each interval and observe that the expression is non-negative on the intervals \([-4, 2]\) and \([8, \infty)\).
  • Final answer: \(\boxed{[-4, 2] \cup [8, \infty)}\)

Explanation

  1. Finding the Roots and Dividing the Number Line

To solve the inequality \((x-8)(x-2)(x+4) \geq 0\), we first identify the points where the product equals zero, which are the roots of the expression \((x-8)(x-2)(x+4) = 0\). Setting each factor to zero, we find the roots:

  • \(x - 8 = 0 \implies x = 8\)
  • \(x - 2 = 0 \implies x = 2\)
  • \(x + 4 = 0 \implies x = -4\)

These roots, \(-4, 2,\) and \(8\), divide the real number line into four distinct intervals: \((-\infty, -4]\), \([-4, 2]\), \([2, 8]\), and \([8, \infty)\).

  1. Testing the Intervals

We now test each interval to determine where the polynomial \(P(x) = (x-8)(x-2)(x+4)\) is greater than or equal to zero.

  • For the interval \((-\infty, -4]\), let's choose \(x = -5\):

\(P(-5) = (-5-8)(-5-2)(-5+4) = (-13)(-7)(-1) = -91\), which is less than 0.

  • For the interval \([-4, 2]\), let's choose \(x = 0\):

\(P(0) = (0-8)(0-2)(0+4) = (-8)(-2)(4) = 64\), which is greater than 0.

  • For the interval \([2, 8]\), let's choose \(x = 3\):

\(P(3) = (3-8)(3-2)(3+4) = (-5)(1)(7) = -35\), which is less than 0.

  • For the interval \([8, \infty)\), let's choose \(x = 9\):

\(P(9) = (9-8)(9-2)(9+4) = (1)(7)(13) = 91\), which is greater than 0.

  1. Determining the Solution Set

Based on the sign analysis, the inequality \((x-8)(x-2)(x+4) \geq 0\) holds when the result is positive or zero. We found this occurs in the intervals \([-4, 2]\) and \([8, \infty)\). Combining these, the solution set is \([-4, 2] \cup [8, \infty)\).

Examples

This method of solving polynomial inequalities is essential in fields like engineering and economics, where you might need to determine the range of variables—such as production levels or load capacities—that keep a system within safe or profitable bounds. By identifying critical thresholds (the roots) and testing the intervals between them, you can clearly define the conditions under which a specific process will be productive. For example, if a company's profit is modeled by a function like \(P(x) = (x-8)(x-2)(x+4)\), they can quickly identify the ranges of production units \(x\) that yield a non-negative profit.

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