Inequalities, regions and feasible choices
Represent limits with inequalities and use test points to identify permitted regions.
What you will tackle
- Create and solve one-variable inequalities, reversing direction when multiplying by a negative
- Model and test two-variable inequalities
- Interpret boundary inclusion and solution regions
- Combine inequalities into feasible systems across algebra, tables and graphs
Direction records the permitted side
Solving −3x + 4 < 13 gives −3x < 9 and then x > −3: division by a negative reverses the order. On a number line, a strict inequality uses an open endpoint; ≤ or ≥ includes the boundary.
A two-variable solution is a region
The boundary of 2x + y ≤ 8 is the solid line y = 8 − 2x. The origin satisfies the inequality, so shade that side. For a system, retain only the overlap of all permitted regions. Nonnegative counts add x ≥ 0 and y ≥ 0; integer restrictions further limit actual choices.
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 −3x + 4 < 13. For 2x + y ≤ 8, test (3,2) and (4,1) and say whether the boundary is included.
Show a starting hint
Substitute both coordinates when testing a point.
Compare with the explained answer
- Divide −3x < 9 by −3 to obtain x > −3.
- At (3,2), 2(3) + 2 = 8, so it is allowed; at (4,1), the sum is 9, so it is not.
- The symbol ≤ includes the boundary, shown by a solid line.
Answer / response criteria: x > −3; (3,2) allowed; (4,1) excluded; boundary included.
Guided 2
Read the shaded feasible-region diagram with solid boundaries through x = 1, y = 0 and the line joining (1,5) to (6,0). Recover the three inequalities.
Compare with the explained answer
- The vertical boundary includes points to its right: x ≥ 1.
- The horizontal boundary includes points above it: y ≥ 0.
- The diagonal is x + y = 6; shading is below it, and the boundary is solid, so x + y ≤ 6.
Answer / response criteria: x ≥ 1, y ≥ 0, x + y ≤ 6.
Independent practice
Independent 1
Find the feasible x values when 2x + 1 ≤ 11 and x > 2. What integer values remain?
Compare with the explained answer
- The first inequality gives x ≤ 5.
- Intersect x ≤ 5 with x > 2.
- The real interval is 2 < x ≤ 5; the integers are 3, 4 and 5.
Answer / response criteria: 2 < x ≤ 5; integers 3, 4, 5.
Independent 2
Use that diagram and its inequalities to test (2,3), (0,2) and (5,2). Explain which condition excludes each rejected point.
Compare with the explained answer
- (2,3) satisfies all three conditions.
- (0,2) fails x ≥ 1.
- (5,2) has x + y = 7 and fails x + y ≤ 6.
Answer / response criteria: Only (2,3) is feasible; the other exclusions are as stated.
Application practice
Application 1
A workshop has at most $60 for kits costing $8 and posters costing $4, and needs at least 9 items. Write the system and test (5 kits, 4 posters) and (6 kits, 3 posters).
Compare with the explained answer
- Let k,p be nonnegative integers. Require 8k + 4p ≤ 60 and k + p ≥ 9.
- For (5,4), cost is 56 and count is 9: feasible.
- For (6,3), cost is 60 and count is 9: feasible; each uses a different point in the overlap.
Answer / response criteria: 8k + 4p ≤ 60, k + p ≥ 9, k,p ≥ 0 integers; both plans are feasible.
Application 2
Interpret the region as a plan for at least one training session x and a nonnegative number of study sessions y, with at most six sessions total. List all permitted integer y values when x = 4, and explain how the graph encodes the decision.
Compare with the explained answer
- Substitute x = 4 into x + y ≤ 6, giving y ≤ 2.
- With nonnegative integer sessions, y is 0,1 or 2.
- These are the integer points on the vertical slice x = 4 inside the shaded region.
Answer / response criteria: y = 0,1,2.
What can you explain now?
Interpret the region as a plan for at least one training session x and a nonnegative number of study sessions y, with at most six sessions total. List all permitted integer y values when x = 4, and explain how the graph encodes the decision.
Compare with the explanation
y = 0,1,2.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
After trying it yourself, choose your next review. This is your self-assessment.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide