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Graphing Inequalities Calculator

Solve and graph linear inequalities ax + b < c, ax + b > c, ax + b ≤ c, or ax + b ≥ c on the number line.

Reviewed by Christopher FloiedPublished Updated

This free online graphing inequalities calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Coefficient of x

Added to ax

Value to compare against

Results

Boundary value

0

Direction flipped (dividing by negative)

0

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Graphing Inequalities Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Graphing Inequalities Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Graphing Inequalities Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Graphing Inequalities Calculator

The Graphing Inequalities Calculator solves and visualizes linear inequalities on the number line. Unlike equations that have exact solutions, inequalities define ranges (intervals) of valid values. Solving ax + b < c involves isolating x, with the critical rule that dividing or multiplying by a negative number reverses the inequality direction. The solution is represented on the number line with open circles (for strict inequalities < and >) or closed/filled circles (for non-strict inequalities ≤ and ≥), with shading extending in the direction of valid values. Inequalities are foundational in optimization, constraint programming, statistical confidence intervals, and real-world applications like budgeting, speed limits, and engineering tolerances.

The Math Behind It

A linear inequality in one variable takes the form ax + b < c (or ≤, >, ≥). To solve, isolate x: subtract b from both sides to get ax < c - b, then divide by a: x < (c - b)/a. The critical rule: if a > 0, the inequality direction is preserved. If a < 0, the inequality direction reverses. This is because multiplying or dividing both sides of an inequality by a negative number reverses the order relation (e.g., 2 < 3 but -2 > -3). The solution set is an interval on the real number line. For x < k, the solution is (-∞, k); for x ≤ k, it is (-∞, k]; for x > k, it is (k, ∞); for x ≥ k, it is [k, ∞). Compound inequalities like a < x < b define bounded intervals. Graphically, open circles indicate excluded endpoints, while closed/filled circles indicate included endpoints. Systems of linear inequalities in two variables define regions in the plane, forming the basis of linear programming. The intersection of multiple inequality constraints is called the feasible region, and optimal solutions occur at vertices of this region (by the fundamental theorem of linear programming).

Formula Reference

Solving Linear Inequality

ax + b < c → x < (c-b)/a (flip if a < 0)

Variables: Divide by a; reverse inequality if a is negative

Worked Examples

Example 1: Simple inequality

Solve 3x + 2 < 11

Step 1:Subtract 2: 3x < 9
Step 2:Divide by 3 (positive, no flip): x < 3
Step 3:Solution: (-∞, 3)
Step 4:Graph: open circle at 3, shade left

x < 3, interval (-∞, 3)

Example 2: Inequality with sign flip

Solve -2x + 5 ≥ 1

Step 1:Subtract 5: -2x ≥ -4
Step 2:Divide by -2 (negative, FLIP the sign): x ≤ 2
Step 3:Solution: (-∞, 2]
Step 4:Graph: closed circle at 2, shade left

x ≤ 2, interval (-∞, 2]

Common Mistakes & Tips

  • !Forgetting to flip the inequality when multiplying or dividing by a negative number
  • !Using a closed circle for strict inequalities (< and >) instead of an open circle
  • !Shading the wrong direction on the number line
  • !Not checking the solution by substituting a test value from the solution set back into the original inequality

Related Concepts

Interval Notation

Express the solution set of an inequality using interval notation

Absolute Value Equation

Absolute value inequalities produce bounded or unbounded intervals

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Absolute Value Equation Calculator

Interval Notation Calculator

Frequently Asked Questions

Why does the inequality flip when dividing by a negative?

The order of real numbers reverses when multiplied by a negative: if a < b, then -a > -b. For example, 2 < 5 but -2 > -5. This reversal applies whenever both sides of an inequality are multiplied or divided by a negative number.

What is the difference between an open and closed circle?

An open circle (○) at a point means the point is NOT included in the solution (strict inequality, < or >). A closed/filled circle (●) means the point IS included (non-strict inequality, ≤ or ≥).

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