Polynomial and rational inequalities
Objective. Solve inequalities with sign diagrams and stated domain restrictions.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A polynomial or rational inequality is solved by finding the critical values where the expression equals zero or is undefined, then using a sign diagram across the resulting intervals to decide where the expression is positive or negative. For (x−1)(x+2)≤0, the critical values are x=1 and x=−2, and testing a point in each region shows the expression is negative only between them, giving −2≤x≤1. For rational inequalities such as (x−3)/(x+1)>0, the denominator's zero (x=−1) must be excluded from the solution even though it is a boundary point. Always state strict or non-strict inequality correctly at each boundary.
Worked example
Solve the inequality (x−2)(x+4) ≤ 0 and express the solution set.
- 1Find critical values from (x−2)(x+4)=0: x=2 and x=−4
- 2Sketch the upward parabola through these roots
- 3Identify the region below the x-axis, which lies between the roots
- 4State the solution as −4 ≤ x ≤ 2
Answer: −4 ≤ x ≤ 2
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Practice
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Start practisingCommon mistake. Students multiply both sides of a rational inequality by the denominator without considering its sign, turning (x−3)/(x+1)>0 into x−3>0 and missing the case where x+1 is negative and the inequality flips.
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