🎯 What You'll Learn in This Lesson
- What it means to graph an inequality in two variables.
- When to use a solid versus dashed boundary line.
- How to choose which side of the line to shade.
- How to solve a SYSTEM of two-variable inequalities (the overlap region).
- Speed tricks for matching graphs to systems.
- 10 trick-based practice questions.
1. What Is a Two-Variable Inequality?
An equation in two variables (like y = 2x + 1) graphs as a single line. An INEQUALITY in two variables (like y ≥ 2x + 1) graphs as an entire REGION — the line plus everything on one side of it.
2. Three Steps to Graph
STEP 1 — Pretend the inequality is an equation. Graph the boundary line.
STEP 2 — SOLID line for ≤ or ≥. DASHED line for < or >.
STEP 3 — Shade the side that satisfies the inequality (use a test point).
3. The Test Point Trick
Pick any point NOT on the line — (0, 0) is usually easiest. Plug it into the original inequality. If the result is TRUE, shade the side containing that point. If FALSE, shade the OTHER side.
📝 Example: Graph y > 2x − 1
Boundary: y = 2x − 1, slope 2, y-intercept −1.
Style: Strict > → DASHED line.
Test (0, 0): 0 > 2(0) − 1 → 0 > −1 ✓ TRUE. Shade the side containing the origin (above the line).
4. Solid vs. Dashed — At a Glance
❓ Does the inequality include the boundary? | ||
≤ or ≥ ▼ | < or > ▼ | |
SOLID line (boundary is part of solution) | DASHED line (boundary is NOT included) | |
5. Systems of Two-Variable Inequalities
Two inequalities graphed together create an OVERLAP region. Any (x, y) point inside the overlap satisfies both inequalities at once.
💡 Identifying the Overlap Region
Shade each inequality lightly with a different color or pattern. The DARKER region (where colors layer) is your solution set.
6. Speed Tricks
⚡ TRICK 1 — Solve for y first
Convert each inequality to y ≤ … or y ≥ … form. Then 'y ≥' means shade ABOVE; 'y ≤' means shade BELOW. Done in 5 seconds.
⚡ TRICK 2 — Origin test
If the boundary doesn't pass through (0, 0), use (0, 0) as your test point. Plug it in: if true, shade the half-plane containing the origin; if false, the other half.
⚡ TRICK 3 — Spot dashed vs. solid in answer choices
On graph-matching multiple-choice questions, the line style (solid/dashed) usually differs across choices. Knock out half the options instantly by checking the inequality symbol.
⚠️ Watch Out!
- Drawing a dashed line for ≤ or ≥, or a solid line for < or >.
- Shading the wrong side because of an arithmetic slip in the test point.
- Forgetting to check that test point is NOT on the boundary line.
- When solving for y, forgetting to flip the inequality if you divided by a negative.
7. Summary
📌 The Whole Topic in 30 Seconds
- Boundary line: Solid for ≤ or ≥; dashed for < or >.
- Shade: Use a test point. (0, 0) is the easiest if not on the line.
- y ≥ form: Shade above the line.
- y ≤ form: Shade below the line.
- System: Solution is the overlap of all shaded regions.
8. Practice — 10 Questions
Q1. Which point is a solution to y < 2x + 1?
A) (0, 5) B) (1, 0) C) (2, 6) D) (3, 8)
Hint: Plug each in. (1, 0): 0 < 3 ✓
Q2. The boundary line of y ≥ 3x − 2 is drawn how?
A) Dashed B) Solid C) Dotted D) None
Hint: ≥ includes the boundary → solid.
Q3. Which inequality is satisfied by (0, 0)?
A) y > 2x + 5 B) y ≤ x − 3 C) y < −x + 1 D) y ≥ 4x + 2
Hint: Plug (0,0): only C gives 0 < 1 ✓
Q4. The graph of y < x has what features?
A) Solid line, shaded above B) Solid line, shaded below C) Dashed line, shaded above D) Dashed line, shaded below
Hint: <: dashed. y < x: shade below.
Q5. Which of the following is NOT a solution to y ≥ −x + 4?
A) (0, 4) B) (5, 5) C) (1, 0) D) (3, 2)
Hint: (1, 0): 0 ≥ 3 is false.
Q6. A system: y > x and y < −x + 6. Which point is in the overlap?
A) (0, 0) B) (4, 1) C) (1, 2) D) (3, 5)
Hint: (1, 2): 2 > 1 ✓ and 2 < 5 ✓
Q7. The boundary of y > 2 is drawn how?
A) Solid horizontal line at y = 2, shaded above B) Dashed horizontal line at y = 2, shaded above C) Dashed vertical line D) Solid vertical line
Hint: y > 2: dashed horizontal, shade above.
Q8. Which inequality describes 'shade below the line y = (1/2)x + 3, boundary included'?
A) y < (1/2)x + 3 B) y ≤ (1/2)x + 3 C) y > (1/2)x + 3 D) y ≥ (1/2)x + 3
Hint: 'Below' = ≤ (boundary included).
Q9. Which point satisfies BOTH 2x + y ≤ 6 AND x − y ≥ 0?
A) (0, 6) B) (3, 0) C) (1, 5) D) (4, 4)
Hint: (3, 0): 6 ≤ 6 ✓ and 3 ≥ 0 ✓
Q10. The line y = −x + 4 is dashed and shaded above. Which inequality matches?
A) y < −x + 4 B) y > −x + 4 C) y ≤ −x + 4 D) y ≥ −x + 4
Hint: Dashed → strict; shaded above → >.
Answer Key & Shortcut Solutions
# | Answer | Shortcut Reason |
Q1. | B) (1, 0) | Plug into y < 2x + 1: 0 < 3 ✓ |
Q2. | B) Solid | ≥ → boundary included → solid line. |
Q3. | C) y < −x + 1 | Origin: 0 < 1 ✓ |
Q4. | D) Dashed, below | < → dashed; y < x → below. |
Q5. | C) (1, 0) | 0 ≥ 3 is false. |
Q6. | C) (1, 2) | Both 2 > 1 and 2 < 5 hold. |
Q7. | B) Dashed, above | y > 2: dashed horizontal, shade above. |
Q8. | B) y ≤ (1/2)x + 3 | Below + included = ≤ |
Q9. | B) (3, 0) | Satisfies both. |
Q10. | B) y > −x + 4 | Strict + above = > |
🚀 You've Got This! 💪
Two-variable inequalities are just one-variable inequalities… in 2D! Master test-point shading and you're set.