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
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