2.7Graphing Linear Inequalities in Two Variables
When an inequality has two variables, the solution is an entire half-plane — a boundary line divides the coordinate plane, and one side satisfies the inequality.
Why This Matters
Two-variable inequalities model real constraints — budget limits, production capacity, nutritional requirements. They're the foundation of linear programming used in business and engineering.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do you determine which half-plane satisfies a two-variable linear inequality, and when is the boundary line solid versus dashed?
Lesson Overview
A linear inequality in two variables (like y < 2x + 1 or 3x + y ≥ 6) has infinitely many solutions — all the points in a region of the coordinate plane called a half-plane. To graph it: (1) Graph the boundary line — use a solid line for ≤ or ≥ (boundary included) and a dashed line for < or > (boundary not included). (2) Choose a test point not on the line — (0, 0) works unless the line passes through the origin. (3) Substitute the test point into the inequality. If it makes the inequality true, shade the side containing the test point. If false, shade the opposite side.
Solid Line — ≤ or ≥
The boundary line is included in the solution. Points on the line satisfy the inequality.
y ≤ 2x + 1 · y ≥ −x + 3
Dashed Line — < or >
The boundary line is not included. Points on the line do NOT satisfy the inequality.
y < 2x + 1 · y > −x + 3
Worked Examples
Graph y < 2x + 1
Boundary line: y = 2x + 1 (slope 2, y-intercept 1).
Strict inequality (<) → draw a DASHED line.
Test point (0,0): 0 < 2(0) + 1 = 1. Is 0 < 1? YES ✓
Shade the side containing (0,0) — below the line.
y < 2x + 1 — dashed line, shade below
Graph y ≥ −x + 3
Boundary line: y = −x + 3 (slope −1, y-intercept 3).
Non-strict inequality (≥) → draw a SOLID line.
Test point (0,0): 0 ≥ −(0) + 3 = 3. Is 0 ≥ 3? NO ✗
Shade the side NOT containing (0,0) — above the line.
y ≥ −x + 3 — solid line, shade above
Graph 2x + y ≤ 4
Rewrite in slope-intercept form: y ≤ −2x + 4.
Boundary line: y = −2x + 4 (slope −2, y-intercept 4).
Non-strict (≤) → SOLID line.
Test point (0,0): 2(0) + 0 ≤ 4 → 0 ≤ 4. YES ✓
Shade the side containing (0,0) — below the line.
Graph x > −2
This is a vertical line at x = −2.
Strict inequality (>) → DASHED vertical line.
Test point (0,0): 0 > −2. YES ✓
Shade the side containing (0,0) — to the right of x = −2.
x > −2 — dashed vertical line, shade right
Graph y ≤ 3
This is a horizontal line at y = 3.
Non-strict (≤) → SOLID horizontal line.
Test point (0,0): 0 ≤ 3. YES ✓
Shade the side containing (0,0) — below y = 3.
Guided Practice
Guided Practice Video: Graphing Linear Inequalities in Two Variables
Watch the guided practice walkthrough for graphing linear inequalities in two variables, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Graph y > x − 3.
Hint: Slope 1, y-intercept −3. Strict > → dashed line. Test (0,0): 0 > −3 ✓. Shade above (origin side).
Graph y ≤ −(1/2)x + 2.
Hint: Slope −1/2, y-intercept 2. Non-strict ≤ → solid line. Test (0,0): 0 ≤ 2 ✓. Shade below (origin side).
Graph 3x − y > 6.
Hint: Rewrite: y < 3x − 6. Dashed line (slope 3, y-int −6). Test (0,0): 0 < −6? NO ✗. Shade opposite side from origin.
Graph x ≤ 4.
Hint: Vertical solid line at x = 4. Test (0,0): 0 ≤ 4 ✓. Shade to the left (origin side).
Graph y ≥ −1.
Hint: Horizontal solid line at y = −1. Test (0,0): 0 ≥ −1 ✓. Shade above (origin side).
Key Vocabulary
Half-Plane
The region on one side of a boundary line in the coordinate plane. A linear inequality in two variables has a half-plane as its solution set.
Example: y > 2x + 1 → all points above the line y = 2x + 1
Boundary Line
The line that separates the coordinate plane into two half-planes. It is the graph of the related linear equation.
Example: For y ≤ 3x − 2, the boundary line is y = 3x − 2.
Solid Line
Used for ≤ or ≥ inequalities. Points on the line ARE solutions.
Example: y ≤ x + 1 → solid line (points on line satisfy y = x + 1 ≤ x + 1)
Dashed Line
Used for < or > inequalities. Points on the line are NOT solutions.
Example: y > x + 1 → dashed line (points on line give y = x + 1, not y > x + 1)
Test Point
A point substituted into the inequality to determine which half-plane to shade. (0,0) is the most convenient unless the line passes through the origin.
Example: Test (0,0) in y < 2x + 1: 0 < 1 ✓ → shade origin's side
Solution Region
The shaded half-plane (and possibly the boundary line) that contains all ordered pairs satisfying the inequality.
Example: For y ≥ −x + 3, the solution region is above the solid line.
Interactive Practice — 5 Questions
When graphing y > 3x − 2, the boundary line should be:
To determine which side of the boundary line to shade, you should:
For y ≤ −2x + 4, which test point confirms the correct shading?
The graph of x ≥ 3 is:
Which inequality is graphed by a solid line with slope 1 and y-intercept −2, shaded above?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Graph y > 3x − 1. Identify: solid or dashed? Which side is shaded?
Graph y ≤ −2x + 5. Identify: solid or dashed? Which side is shaded?
Graph 4x + 2y ≥ 8. Rewrite in slope-intercept form first.
Graph x < −1. Identify: solid or dashed? Which side is shaded?
A student has at most $40 to spend on notebooks (n) at $3 each and pens (p) at $2 each. Write and graph the inequality 3n + 2p ≤ 40. What does the shaded region represent?
ChallengeCommon Mistakes
Using a solid line for a strict inequality (< or >) — including the boundary when it should be excluded.
Strict inequalities (< or >) use a DASHED line. Non-strict (≤ or ≥) use a SOLID line. The symbol tells you: if there's an 'equal to' part, the line is solid.
Shading the wrong side — shading above when the inequality says below, or vice versa.
Always use a test point to confirm. Substitute (0,0) into the original inequality. If true, shade the origin's side. If false, shade the opposite side.
Forgetting to rewrite the inequality in slope-intercept form before graphing — misidentifying the slope or y-intercept.
Rewrite ax + by ≤ c as y ≤ (−a/b)x + (c/b) first. This makes it easy to identify slope and y-intercept for graphing.
Using (0,0) as a test point when the boundary line passes through the origin.
If the line passes through (0,0), choose a different test point like (1,0) or (0,1) to avoid substituting a point that lies on the boundary.
Math Tips
Dashed = strict: if the inequality symbol has no "equal to" (< or >), the line is dashed. If it does (≤ or ≥), the line is solid.
Test point shortcut: (0,0) works for most problems. Just substitute x=0 and y=0 into the original inequality and check if it is true or false.
Shade direction for y-form: if the inequality is y > or y ≥, shade ABOVE the line. If y < or y ≤, shade BELOW. This only works when the inequality is solved for y.
SAT/ACT tip: to check if a point is in the solution region, substitute it into the inequality. If it satisfies the inequality, it is in the shaded region.