Write Inequality From Graph Calculator – With Steps, Graph & Number Line (Free)
Describe what you see on a graph — slope, y-intercept, line type, and shading direction — and get the inequality written out for you with full step-by-step explanation. Or type any linear inequality and graph it instantly. Free, online, no app needed.
Look at your graph. Read off the slope, y-intercept, whether the boundary line is solid or dashed, and which side is shaded — then enter them below.
How to Write an Inequality From a Graph — Step by Step
Reading an inequality from a graph is a core algebra skill, but it trips up a surprising number of students because it combines several observations at once: the slope and intercept of the boundary line, the type of line, and the direction of shading. Break the problem into four clear steps and it becomes mechanical.
Step 1 — Find the boundary line equation. Identify two clear points on the boundary line and calculate the slope (rise ÷ run). Then find the y-intercept (where the line crosses the y-axis). Write the line as y = mx + b.
Step 2 — Identify the line type. Is the boundary line solid or dashed? Solid → the boundary is included (use ≤ or ≥). Dashed → the boundary is excluded (use < or >).
Step 3 — Identify the shaded region. Which side of the boundary line is shaded? Shading above the line means the solution is y > or y ≥. Shading below means y < or y ≤.
Step 4 — Combine and verify. Write the full inequality (e.g. y > 2x + 3). Then test a point in the shaded region — substitute its coordinates and confirm the inequality is satisfied. The origin (0, 0) is the easiest test point as long as the boundary doesn't pass through it.
Solid Line vs Dashed Line: The Most Common Mistake
Confusing solid and dashed lines is the single most frequent error when writing an inequality from a graph. The rule is simple but must become automatic:
- Dashed line → strict inequality: use < or >. The boundary itself is NOT part of the solution. Students sometimes describe this as "the line doesn't count."
- Solid line → non-strict inequality: use ≤ or ≥. The boundary IS part of the solution — every point on the line satisfies the inequality.
Memory trick: think of a dashed line as a dotted boundary you can't actually touch — strict symbols, no equal sign. A solid line is one you can stand on — it includes the equal-to condition.
Graphing Linear Inequalities Calculator: How It Works
When you use the graphing linear inequalities tab above, the tool takes your inequality in slope-intercept form (y > mx + b, y ≤ mx + b, etc.) and: finds the slope and y-intercept, draws the boundary line (solid or dashed based on your symbol), tests the origin to confirm shading direction, fills the correct half-plane, and shows every step of the reasoning. The visual output matches exactly what a textbook graph would show — making it ideal for checking homework or exam practice.
Reading Slope and Y-Intercept From a Graph
To write an inequality from a graph, you need the boundary line's equation. Here's how to reliably extract slope and y-intercept from a plotted line:
- Y-intercept (b): Find where the line crosses the y-axis. If it crosses at y = 3, then b = 3. This is usually the easier number to read directly.
- Slope (m): Pick two points on the line that land exactly on grid intersections. Count the vertical change (rise) and horizontal change (run) between them. Slope = rise ÷ run. If you move right 2 and up 4, slope = 4 ÷ 2 = 2. If you move right 3 and down 1, slope = −1 ÷ 3 = −1/3.
- Write the line: Substitute m and b into y = mx + b.
- Add the inequality symbol: Use the line type (solid/dashed) and shading direction.
Inequality Number Line Calculator: One-Variable Inequalities
Not all inequalities involve two variables and a coordinate plane. Single-variable inequalities — like x > 3 or x ≤ −2 — are shown on a number line rather than a coordinate graph. The rules for reading and writing them follow a parallel logic:
| What you see on the number line | What it means | Inequality |
|---|---|---|
| Open circle at 3, arrow pointing right | x is greater than 3, not including 3 | x > 3 |
| Filled circle at 3, arrow pointing right | x is greater than or equal to 3 | x ≥ 3 |
| Open circle at −2, arrow pointing left | x is less than −2, not including −2 | x < −2 |
| Filled circle at −2, arrow pointing left | x is less than or equal to −2 | x ≤ −2 |
| Filled circle at 1, filled circle at 5, shaded between | x is between 1 and 5, inclusive | 1 ≤ x ≤ 5 |
The number line calculator tab above handles all of these forms — enter an inequality like x > 3 or x <= -2 and the tool draws the correct number line automatically.
Common Examples: Write Inequality From Graph
| Slope | Y-Intercept | Line Type | Shading | Inequality |
|---|---|---|---|---|
| 2 | 3 | Dashed | Above | y > 2x + 3 |
| −1 | 4 | Solid | Below | y ≤ −x + 4 |
| 0.5 | −1 | Solid | Above | y ≥ 0.5x − 1 |
| 1 | 0 | Dashed | Below | y < x |
| −2 | 5 | Solid | Below | y ≤ −2x + 5 |
| 0 | 3 | Dashed | Above | y > 3 (horizontal line) |
| 1/3 | −2 | Solid | Above | y ≥ (1/3)x − 2 |
System of Inequalities: When Two or More Inequalities Overlap
A system of inequalities involves two or more inequalities graphed on the same coordinate plane. The solution is the region where all shaded areas overlap — called the intersection or feasible region. To identify it: graph each inequality separately, shade each one, and then identify the region that is shaded by every inequality at once. Any point in that overlap region satisfies all inequalities in the system simultaneously.
Reading a system of inequalities from a graph follows the same process as reading a single inequality — just apply it separately to each boundary line, then observe which region is shaded all at once (not just by one inequality). The boundaries of the feasible region form a polygon in linear programming problems, and the corner points of this polygon are the critical points for optimization.
Frequently Asked Questions
Q: How do you write an inequality from a graph?
A: Find the slope and y-intercept of the boundary line, check whether it's solid (≤, ≥) or dashed (<, >), identify which side is shaded (above → > or ≥; below → < or ≤), then combine. Always verify with a test point from the shaded region.
Q: What does a solid line mean in an inequality graph?
A: A solid boundary line means the line itself is included in the solution set — the inequality uses ≤ or ≥. A dashed line means the boundary is excluded, so the inequality uses strict < or >.
Q: How do you graph a linear inequality?
A: Write the inequality in slope-intercept form, graph the boundary line (solid for ≤/≥, dashed for </>), test the origin in the inequality, and shade the correct half-plane. The shaded region is the complete solution set.
Q: How do you graph an inequality on a number line?
A: Draw an open circle at the boundary value for < or > (not included), or a filled circle for ≤ or ≥ (included). Then shade an arrow to the left for "less than" or to the right for "greater than."
Q: How do you solve an inequality and graph the solution?
A: Isolate the variable using the same rules as equations — but if you multiply or divide both sides by a negative number, flip the inequality sign. Once solved, graph the solution on a number line (one variable) or shade the half-plane (two variables).
Q: What is a system of inequalities?
A: A system of inequalities is two or more inequalities graphed together. The solution is the overlapping shaded region that satisfies all inequalities at the same time. Each boundary line is identified and shaded independently, and the feasible region is where all shaded areas intersect.