Inequalities & Graphing
Graphing Linear Inequalities
Graph a linear inequality by first graphing the boundary line (solid for ≤ or ≥, dashed for < or >), then shading the region that satisfies the inequality.
Visualize It
Boundary line + shaded region
Watch & Learn
Watch a clear, friendly video explanation of Graphing Linear Inequalities:
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Graph y > 2x − 1
Step-by-Step Solution
Graph the boundary line y = 2x − 1 as a DASHED line (> means not equal).
Test point (0, 0): Is 0 > 2(0) − 1? Is 0 > −1? YES.
Shade the region containing (0, 0) — above the line.
Answer
Dashed line y = 2x − 1, shade above
y > 2x − 1 (shade above dashed line)
Problem
Graph y ≤ −x + 4
Step-by-Step Solution
Graph y = −x + 4 as a SOLID line (≤ means equal is included).
Test (0, 0): Is 0 ≤ −0 + 4? Is 0 ≤ 4? YES.
Shade the region containing (0, 0) — below the line.
Answer
Solid line y = −x + 4, shade below
y ≤ −x + 4 (shade below solid line)
Problem
Graph y ≥ 3
Step-by-Step Solution
Graph the horizontal line y = 3 as a SOLID line.
Shade above the line (all y values ≥ 3).
Answer
Solid horizontal line at y = 3, shade above
y ≥ 3 (shade above y = 3)
Problem
Graph y < x
Step-by-Step Solution
Graph y = x as a DASHED line (strict inequality).
Test (1, 0): Is 0 < 1? YES.
Shade the region below the line containing (1, 0).
Answer
Dashed line y = x, shade below
y < x (shade below dashed line)
Problem
Graph y ≥ −2x + 1
Step-by-Step Solution
Graph y = −2x + 1 as a SOLID line (≥ includes equal).
Test (0, 0): Is 0 ≥ −2(0) + 1? Is 0 ≥ 1? NO.
Shade the region NOT containing (0, 0) — above the line.
Answer
Solid line y = −2x + 1, shade above
y ≥ −2x + 1 (shade above solid line)
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Is the boundary line solid or dashed for y < x + 2?
Is the boundary line solid or dashed for y ≥ 3x?
Test (0,0) in y > x − 5. Is it in the solution region?
For y ≤ 2x + 1, do you shade above or below the line?
For y > −2x + 3, do you shade above or below?
Test (2, 5) in y > 2x − 1. Is it in the solution region?
Is the boundary line solid or dashed for y ≤ −x?
For y < x + 4, test (0, 0). Is it a solution?
For y > 3x − 2, test (0, 0). Is it a solution?
For y ≤ x − 5, test (0, 0). Is it a solution?
What is the first step when graphing a linear inequality?
For y ≥ 2x + 1, do you shade above or below?
Finished all 12? Give yourself a pat on the back — then check your work!
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