Questions: Solve the inequality. (Enter your answer using interval notation.) x^2 + 4x / x^2 - 25 ≤ 0 Graph the solution set.

Solve the inequality. (Enter your answer using interval notation.) x^2 + 4x / x^2 - 25 ≤ 0 Graph the solution set.
Transcript text: Solve the inequality. (Enter your answer using interval notation.) \[ \frac{x^{2}+4 x}{x^{2}-25} \leq 0 \] Graph the solution set.
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Solution

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Solution Steps

Step 1: Simplify the inequality

We are asked to solve the inequality:

(x² + 4x) / (x² - 25) ≤ 0

Factor the numerator and denominator:

[x(x+4)] / [(x-5)(x+5)] ≤ 0

Step 2: Find the critical points

The critical points are the values of x where the numerator or denominator equals zero. These are x = 0, x = -4, x = 5, and x = -5.

Step 3: Test intervals

We test the sign of the expression in each interval determined by the critical points:

  • x < -5: Choose x = -6. The expression becomes [(-6)(-2)] / [(-11)(-1)] = 12 > 0.
  • -5 < x < -4: Choose x = -4.5. The expression becomes [(-4.5)(-0.5)] / [(-9.5)(-0.5)] = 2.25 / 4.75 > 0.
  • -4 < x < 0: Choose x = -1. The expression becomes [(-1)(3)] / [(-6)(4)] = -3 / -24 > 0.
  • 0 < x < 5: Choose x = 1. The expression becomes [(1)(5)] / [(-4)(6)] = 5 / -24 < 0.
  • x > 5: Choose x = 6. The expression becomes [(6)(10)] / [(1)(11)] = 60/11 > 0.
Step 4: Determine the solution set

The inequality is satisfied when (x² + 4x) / (x² - 25) is less than or equal to 0. From our analysis, this happens in the interval (0, 5). The inequality is also satisfied at x = 0 and x=-4 (where the expression evaluates to 0), but not at x = 5 or x = -5, where the denominator becomes 0 and the expression is undefined.

Therefore, the solution set includes x = -4 and x = 0, as well as the interval where the expression is strictly negative (0,5). This is the interval (-4,0) ∪ (0,5) which can be written more concisely as (-4, 0] ∪ (0,5).

Final Answer:

[-4, 0] ∪ (0, 5)

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