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Find the solution to the inequality: x^2 - 9 > 0.

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Question: Find the solution to the inequality: x^2 - 9 > 0.

Options:

  1. (-∞, -3) ∪ (3, ∞)
  2. (-3, 3)
  3. (-3, ∞)
  4. (-∞, 3)

Correct Answer: (-∞, -3) ∪ (3, ∞)

Solution: 

Step 1: Factor the inequality: (x - 3)(x + 3) > 0. Step 2: The critical points are x = -3 and x = 3. Step 3: Test intervals: The solution set is (-∞, -3) ∪ (3, ∞).

Find the solution to the inequality: x^2 - 9 > 0.

Practice Questions

Q1
Find the solution to the inequality: x^2 - 9 > 0.
  1. (-∞, -3) ∪ (3, ∞)
  2. (-3, 3)
  3. (-3, ∞)
  4. (-∞, 3)

Questions & Step-by-Step Solutions

Find the solution to the inequality: x^2 - 9 > 0.
  • Step 1: Start with the inequality x^2 - 9 > 0. This can be factored into (x - 3)(x + 3) > 0.
  • Step 2: Identify the critical points where the expression equals zero. Set (x - 3)(x + 3) = 0. This gives us x = -3 and x = 3.
  • Step 3: These critical points divide the number line into intervals. We will test the intervals: (-∞, -3), (-3, 3), and (3, ∞).
  • Step 4: Choose a test point from each interval and substitute it into the factored inequality (x - 3)(x + 3) > 0 to see if the inequality holds.
  • Step 5: For the interval (-∞, -3), choose x = -4: (-4 - 3)(-4 + 3) = (-7)(-1) > 0, so this interval is part of the solution.
  • Step 6: For the interval (-3, 3), choose x = 0: (0 - 3)(0 + 3) = (-3)(3) < 0, so this interval is NOT part of the solution.
  • Step 7: For the interval (3, ∞), choose x = 4: (4 - 3)(4 + 3) = (1)(7) > 0, so this interval is part of the solution.
  • Step 8: Combine the intervals where the inequality holds true. The solution set is (-∞, -3) ∪ (3, ∞).
  • Inequalities – Understanding how to solve quadratic inequalities by factoring and testing intervals.
  • Critical Points – Identifying points where the expression equals zero to determine intervals for testing.
  • Interval Testing – Using test points in each interval to determine where the inequality holds true.
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