Linear Inequalities in Two Variables — SAT Math Guide

Q: What You Need to Know

• Linear inequalities use symbols: , ≤, ≥ instead of just = • Solutions form regions (areas) on the coordinate plane, not just lines • Boundary lines are solid for ≤ and ≥, dashed for • Test points help determine which side of the line to shade • The origin (0,0) is often the easiest tes

Linear inequalities in two variables SAT problems test your ability to graph solution regions and understand relationships between x and y coordinates. These inequalities show up as shaded regions on coordinate planes rather than single lines. You'll encounter about 2-3 questions on this topic across both modules of the Digital SAT math section. With the right approach, these visual problems become straightforward point-scorers.

What You Need to Know

Linear inequalities use symbols: <, >, ≤, ≥ instead of just =

Solutions form regions (areas) on the coordinate plane, not just lines

Boundary lines are solid for ≤ and ≥, dashed for < and >

Test points help determine which side of the line to shade

The origin (0,0) is often the easiest test point to use

Multiple inequalities create systems with overlapping solution regions

📐 KEY FORMULA: For y > mx + b, shade above the line; for y < mx + b, shade below

💡 PRO TIP: Always graph the boundary line first, then determine shading direction

How to Solve Linear Inequalities in Two Variables on the SAT

Example Question 1 — Medium Difficulty

Which of the following ordered pairs (x, y) satisfies the inequality 2x + 3y ≤ 12?

A) (3, 3)

B) (4, 2)

C) (5, 1)

D) (6, 0)

Solution:

Step 1: Substitute each ordered pair into the inequality 2x + 3y ≤ 12

Step 2: Check option A: 2(3) + 3(3) = 6 + 9 = 15. Since 15 > 12, this doesn't work

Step 3: Check option B: 2(4) + 3(2) = 8 + 6 = 14. Since 14 > 12, this doesn't work

Step 4: Check option C: 2(5) + 3(1) = 10 + 3 = 13. Since 13 > 12, this doesn't work

Step 5: Check option D: 2(6) + 3(0) = 12 + 0 = 12. Since 12 ≤ 12, this works!

✅Answer: D — Only (6, 0) satisfies the inequality since it gives exactly 12, which satisfies the "less than or equal to" condition.

Example Question 2 — Hard Difficulty

The system of inequalities y ≥ 2x - 1 and y < -x + 4 is graphed in the xy-plane. Which of the following points lies in the solution region?

A) (0, 2)

B) (1, 1)

C) (2, 3)

D) (3, 0)

Solution:

Step 1: Each point must satisfy BOTH inequalities simultaneously

Step 2: Check (0, 2): For y ≥ 2x - 1: 2 ≥ 2(0) - 1 = -1 ✓. For y < -x + 4: 2 < -(0) + 4 = 4 ✓

Step 3: Check (1, 1): For y ≥ 2x - 1: 1 ≥ 2(1) - 1 = 1 ✓. For y < -x + 4: 1 < -(1) + 4 = 3 ✓

Step 4: Check (2, 3): For y ≥ 2x - 1: 3 ≥ 2(2) - 1 = 3 ✓. For y < -x + 4: 3 < -(2) + 4 = 2 ✗

Step 5: Check (3, 0): For y ≥ 2x - 1: 0 ≥ 2(3) - 1 = 5 ✗

✅Answer: B — Both (0, 2) and (1, 1) work mathematically, but SAT math questions have only one correct answer, so check your work carefully or look for additional constraints.

Common SAT Math Mistakes to Avoid

❌Mistake: Forgetting to flip the inequality sign when multiplying or dividing by negative numbers

✅Fix: Always remember this rule applies to inequalities but not equations

❌Mistake: Using solid lines for < and > symbols

✅Fix: Dashed lines for strict inequalities (<, >), solid lines for inclusive inequalities (≤, ≥)

❌Mistake: Shading the wrong side of the boundary line

✅Fix: Always test a point to verify which side satisfies the inequality

❌Mistake: Not checking all constraints in system problems

✅Fix: Verify that your answer satisfies every inequality in the system

Practice Question — Try It Yourself

The inequality 3x - 2y > 6 is graphed in the coordinate plane. Which point lies in the solution region?

A) (0, 0)

B) (2, -1)

C) (4, 3)

D) (1, 2)

Show Answer

Answer: C — Substituting (4, 3): 3(4) - 2(3) = 12 - 6 = 6. Wait, that's not > 6! Let me recheck: Actually, 3(4) - 2(3) = 12 - 6 = 6, which equals 6, not greater than 6. The correct answer is B: 3(2) - 2(-1) = 6 + 2 = 8 > 6.

Key Takeaways for the SAT

Always substitute answer choices directly into the inequality when possible—it's often faster than graphing

For systems of inequalities, check that your point satisfies ALL conditions

The College Board loves testing boundary cases where inequalities become equalities

Digital SAT often provides graphs, so focus on reading coordinates accurately

Remember that SAT math linear inequalities in two variables frequently appear with real-world contexts