SAT Linear inequalities
Solve, model, or test points against inequality constraints.
How to score it
- To test a point, plug in and check the inequality holds — flip the sign only when multiplying/dividing by a negative.
- “At least” → ≥, “at most” → ≤, “no more than” → ≤.
- For max/min, set the binding constraint to equality and solve.
Common traps
- Compound “at least … and at most …” modeled with one inequality.
- Forgets to flip the inequality when dividing by a negative.
- Solid vs. dashed boundary line confused on graph items.
The 6 question types, with real examples
Is this point a solution?
“Which point (x, y) is a solution to the given inequality?”
Which point (x, y) is a solution to the given inequality in the xy-plane?
- A
- B
- C✓
- D
Substitute (3, 2) into to get 3 + 5(2) < 15, which simplifies to 13 < 15. This is true, so (3, 2) is a solution.
Max / min from a constraint
“What is the maximum number of [item] that … ?”
A factory produces chairs and tables. Each chair requires 3 hours of labor and each table requires 5 hours of labor. The factory has at most 240 hours of labor available per week. Additionally, the factory must produce at least 20 chairs per week to meet customer demand. What is the maximum number of tables the factory can produce in one week?
- A
- B
- C✓
- D
Let c represent chairs and t represent tables. The constraints are and . To maximize t, minimize c by setting . Substituting: , so , giving and . The maximum number of tables is 36.
Write the modeling inequality
“Which inequality shows this relationship?”
A caterer charges $450 for setup plus $18 per person for a dinner event. An organization has a budget of at most $1800 for the event. Which inequality represents all possible values of p, the number of people who can attend?
- A
- B
- C✓
- D
The total cost is , which must be at most 1800. This gives . Subtracting 450 yields , and dividing by 18 gives .
Compound inequality / 'at most X less than Y'
“A number [n] is at most [c1] less than [c2] times the value of [other]. If [other] is [v], what is the greatest possible value of [n]?”
A number k is at most 5 less than 4 times the value of m. If m is 9, what is the greatest possible value of k?
- A
- B
- C✓
- D
The phrase 'at most 5 less than 4 times the value of m' translates to . Substituting gives (9) - 5 = 36 - 5 = 31. The greatest possible value of k is 31.
Graph of inequality boundary
“[FIGURE: line with shaded region]. Which inequality represents the graph?”
In the xy-plane, a line has slope 4 and y-intercept -5. The region above the line, including the line itself, is shaded. Which inequality represents the graph?
- A✓
- B
- C
- D
With slope 4 and y-intercept -5, the line equation is . Since the region above is shaded and the line is solid, the inequality is .
Solve / rewrite a one-variable linear inequality
“Which inequality is equivalent to [INEQUALITY]? (or: What is the solution to [INEQUALITY]?)”
Which inequality is equivalent to ?
- A
- B✓
- C
- D
Subtracting 8 from both sides gives . Dividing both sides by -4 and flipping the inequality sign gives .
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