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Polynomial and Rational Inequalities

A free College Algebra lesson from the “Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.

To solve f(x) > 0 (or <, >=, <=), find the values where f(x) = 0 (and where f(x) is undefined, for rational expressions). Those critical values split the number line into intervals. Test a point in each interval: if f is positive there, the whole interval satisfies > 0. For rational inequalities, denominator zeros are always EXCLUDED from the solution (never include).

What you'll learn

Why it matters: Profitability ranges in business (where is profit positive?), structural-load tolerance windows, and any 'safe / unsafe' modeling using a polynomial or rational formula leans on this technique.

Worked example

Problem. Solve (x - 1)(x - 3) > 0. Enter the upper boundary of the leftmost solution interval.

  1. Zeros at x = 1 and x = 3.
  2. Test intervals: x < 1 (positive), 1 < x < 3 (negative), x > 3 (positive).
  3. Solution: x < 1 OR x > 3. The leftmost interval ends at x = 1.

Answer: 1

Practice problems

1. Solve x^2 - 9 > 0. Enter the smaller boundary of the solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 - 9 > 0. Enter the smaller boundary of the solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x^2 - 9 = (x - 3)(x + 3). Zeros at -3 and 3.
  4. Solution x < -3 or x > 3. Smaller boundary is -3.
  5. Check the result by substituting or estimating: the response should match -3 and make sense in the original problem.

Answer: -3

2. Same inequality. Enter the larger boundary.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Same inequality. Enter the larger boundary.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Larger boundary is 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Solve x^2 < 16. Enter the upper boundary of the solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 < 16. Enter the upper boundary of the solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x^2 - 16 < 0 -> (x - 4)(x + 4) < 0.
  4. Solution -4 < x < 4. Upper boundary 4.
  5. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. Solve (x - 1)(x - 3) > 0. Enter the LOWER boundary of the rightmost interval.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve (x - 1)(x - 3) > 0. Enter the LOWER boundary of the rightmost interval.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Zeros at 1 and 3. Solution x < 1 or x > 3.
  4. Rightmost interval starts at x = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. Solve x^2 - 5x + 6 <= 0. Enter the smaller boundary.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x^2 - 5x + 6 <= 0. Enter the smaller boundary.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor: (x - 2)(x - 3) <= 0.
  4. Solution 2 <= x <= 3. Smaller boundary 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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