Compound, absolute-value and quadratic inequalities
Represent solution sets with intervals and number-line endpoints, then test the signs that make an inequality true.
What you will tackle
- Read open and closed number-line endpoints
- Solve compound and absolute-value inequalities
- Use unions for separated intervals
- Solve a quadratic inequality with a sign chart
Intervals describe sets
A closed endpoint includes its value; an open endpoint excludes it. The interval [a,b) means a ≤ x < b. An ‘and’ condition takes an intersection, while ‘or’ takes a union. Multiplying an inequality by a negative number reverses its direction.
Distance and signs
For c ≥ 0, |u| ≤ c means −c ≤ u ≤ c, while |u| > c means u < −c or u > c. For a product such as (x − 1)(x − 4), roots divide the line into sign intervals. Test one point in each interval and include roots only when the inequality allows equality.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Solve |2x − 3| ≤ 5 and describe its number-line plot. Solve (x − 1)(x − 4) ≤ 0.
Show a starting hint
For the quadratic, test values 0, 2 and 5.
Compare with the explained answer
- −5 ≤ 2x − 3 ≤ 5 gives −2 ≤ 2x ≤ 8, hence −1 ≤ x ≤ 4.
- Shade the segment between closed endpoints −1 and 4.
- The quadratic product is positive outside its roots and negative between them; include both roots to obtain [1,4].
Answer / response criteria: [−1,4] with closed endpoints; [1,4].
Independent practice
Independent 1
Solve |x + 2| > 3 and (x + 2)(x − 3) > 0. Solve the compound inequality −1 < 2x + 1 ≤ 7.
Compare with the explained answer
- x + 2 < −3 or x + 2 > 3 gives x < −5 or x > 1.
- The product is positive outside roots −2 and 3, with both roots excluded.
- Subtract 1 throughout and divide by 2: −1 < x ≤ 3.
Answer / response criteria: (−∞,−5) ∪ (1,∞); (−∞,−2) ∪ (3,∞); (−1,3].
Application practice
Application 1
A model declares an acceptable setting when |t − 20| ≤ 2 and (t − 19)(t − 21) ≥ 0. Find all acceptable real settings and explain the endpoints.
Compare with the explained answer
- The distance condition gives 18 ≤ t ≤ 22.
- The product condition gives t ≤ 19 or t ≥ 21, including both zeros.
- Their intersection is [18,19] ∪ [21,22]; every endpoint satisfies both non-strict inequalities.
Answer / response criteria: [18,19] ∪ [21,22].
What can you explain now?
A model declares an acceptable setting when |t − 20| ≤ 2 and (t − 19)(t − 21) ≥ 0. Find all acceptable real settings and explain the endpoints.
Compare with the explanation
[18,19] ∪ [21,22].
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide