Solving inequalities
By the end, you can
- Solve rational inequalities with a sign diagram
- Distinguish zeros from excluded denominator values
- Interpret absolute values as distances
about 15 minutes for the lesson · practise at your own pace.
The central idea
An inequality asks for a set of inputs. Factor a rational expression, mark every numerator zero and denominator zero, and test the intervals between them. For , means , while selects the two outside intervals.
Worked example
Solve . Critical inputs are and . The expression is positive when both factors have the same sign: or . The denominator zero is excluded, and the strict inequality excludes the numerator zero.
Multiplying through by a denominator of unknown sign can reverse the answer on only part of the domain.
Check your understanding
Solve .
, hence . Both endpoints are included.
Mastery learning objectives
At the end of this chapter, I should be able to:
Choose a practice question
Use the examples to learn a method, then attempt the linked questions before opening their working. The skill routes below retain the source’s part references.
| Skill | Lecture examples | Discussion | Review follow-up |
|---|---|---|---|
| Analytical: rational functions | 4 | 1(a)(c)(d), 4, 8 first part | 1 first part, 2 first two parts, 3(i), 4 first part |
| Analytical: completing the square | 5, 6, 10 first part | 1(b), 7 | 5, 6, 7 first part, 8(a), 9 first part |
| Analytical: modulus | 8 | 2 | — |
| Graphical methods | 6, 7, 9, 10 first part | 3 first part | 10, 11, 12, 13, 14(i) |
| Hence: modulus replacement | 10 hence | 4(ii) | 1 hence, 7 hence |
| Hence: logarithmic or exponential replacement | 11 hence | 3 hence, 4(i) | 3(ii), 4 hence, 14(ii) |
| Hence: trigonometric replacement | — | 8 hence | — |
| Hence: algebraic replacement | — | — | 2 deduce, 8(b), 9 deduce |
| Prior topic: graphing techniques | — | 5, 6 | 15 |
The printed logarithm row names Example 10 (hence), but that example uses modulus. Example 11 is the logarithm example, so its link is used here. The graphical row also lists Example 10’s first part: its printed working uses a sign test; you can practise a graph of the same expression as an alternative.
Practice questions
38 examples and practice questions, in 7 sections.
More practice
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