3.7 Polynomial and Rational Inequalities
Ideas in this lesson
- Solve factored and expanded polynomial inequalities
- Handle multiplicities, irreducible factors and polynomial substitution
- Solve rational sign inequalities including irrational cut points
- Solve nonzero-side and multi-fraction inequalities
- Compare two graphs through their difference
- Find even-root and reciprocal-root domains
- Solve inequalities graphically and verify rounded boundaries
- Solve inequalities with fractional powers and radical denominators
- Solve parameterized inequalities by ordered cut points
- Model threshold intervals with physical and integer restrictions
Solve factored and expanded polynomial inequalities · Worked example
Move all terms to one side and solve \(x^{3} + 5 x^{2} - 11 x - 80\le5 x\). Factor completely over the reals and show a sign table. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(\left(-\infty, -5\right] \cup \left[-4, 4\right]\)
numerator factor \(x - 4\), signs from left to right: negative, negative, negative, positive.
numerator factor \(x + 4\), signs from left to right: negative, negative, positive, positive.
numerator factor \(x + 5\), signs from left to right: negative, positive, positive, positive.
With zero on the right, factor the expression as \(\left(x - 4\right) \left(x + 4\right) \left(x + 5\right)\).
Cut points: \(-5\), \(-4\), \(4\). Original excluded inputs: none.
Sign chart: between \(-\infty\) and \(-5\): negative; between \(-5\) and \(-4\): positive; between \(-4\) and \(4\): negative; between \(4\) and \(\infty\): positive.
Include numerator zeros that are in the original domain; never include denominator zeros.
The solution is \(\left(-\infty, -5\right] \cup \left[-4, 4\right]\).
Written-work review:
numerator factor \(x + 4\), signs from left to right: negative, negative, positive, positive.
numerator factor \(x + 5\), signs from left to right: negative, positive, positive, positive.
With zero on the right, factor the expression as \(\left(x - 4\right) \left(x + 4\right) \left(x + 5\right)\).
Cut points: \(-5\), \(-4\), \(4\). Original excluded inputs: none.
Sign chart: between \(-\infty\) and \(-5\): negative; between \(-5\) and \(-4\): positive; between \(-4\) and \(4\): negative; between \(4\) and \(\infty\): positive.
Include numerator zeros that are in the original domain; never include denominator zeros.
The solution is \(\left(-\infty, -5\right] \cup \left[-4, 4\right]\).
Written-work review:
- Show the stated algebra and preserve every excluded input.
- Include all requested graph, sign-table, model or explanation components.
- Compare the written work with the supplied reference solution; selecting a statement alone does not establish mastery.