Expressions & Equations

Writing & Graphing Inequalities

Writing an inequality to represent a constraint and graphing its solutions.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Solving One-Variable Equations

When More Than One Answer Is Correct

You have $20 to spend at an arcade, and each game costs $2. Before reading on, think about this: is there exactly one number of games you could play, or many possible correct answers? How is this situation different from an equation like 2g = 20?

Definition — Inequality

A mathematical statement comparing two expressions using <, >, ≤, or ≥, rather than an equal sign — true for a whole range of values rather than just one.

The arcade situation isn't asking for an exact amount spent — it's asking for a limit that can't be crossed. That's exactly what an inequality like 2g ≤ 20 captures: playing 10 games spends exactly the limit, but playing 0 through 9 games also satisfies the constraint. An inequality's solution isn't one number; it's every number that keeps the statement true.

Worked Example — Writing an Inequality from a Constraint

A elevator has a weight limit of 2,000 pounds. Write an inequality for w, the total weight it can safely hold. Since the total weight must stay at or below the limit: w ≤ 2,000.

Worked Example — Graphing an Inequality on a Number Line

Graph the solutions to x > 3 on a number line. Since 3 itself doesn't satisfy 'greater than 3,' mark 3 with an open circle. Since every number greater than 3 does satisfy it, shade the number line to the right of 3, continuing infinitely.

Tip

Use an open circle for < or > (the boundary value itself isn't a solution) and a filled circle for ≤ or ≥ (the boundary value is included) — a quick visual check for which symbol was actually used.

Common Mistakes

Using a filled circle for a strict inequality like x > 3, incorrectly including 3 itself as a solution.
Check the symbol carefully: < and > use an open circle since the boundary number itself doesn't satisfy the inequality; ≤ and ≥ use a filled circle since it does.
Treating an inequality's solution as a single number, the way an equation's solution usually is.
An inequality is satisfied by an entire range of numbers, not just one — shading a ray on the number line, rather than marking a single point, represents every value that works.

Key Takeaways

An inequality compares two expressions using <, >, ≤, or ≥, and is satisfied by a whole range of values, not just one.
Graphing an inequality shades every value that makes it true, using an open circle for a strict inequality and a filled circle otherwise.
Many real-world constraints, like weight limits or budgets, are naturally described by inequalities rather than equations.

Summary

Inequalities represent constraints with a whole range of valid solutions, graphed as a shaded ray on the number line. The next unit shifts from algebra to geometry, finding the area of shapes beyond simple rectangles.

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