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.
Solve and graph a linear inequality
ax + b ≥ c
x ≤ 2
Interval notation: (−∞, 2]
Subtract b, then divide by a and reverse the inequality because a is negative. The endpoint is included.
Shared inputs are stored in the link fragment. Anyone receiving the link can read them.
Learn with a worked scenario
One-variable linear inequalities ax + b compared with c, drawn on a number line.
Question
Solve −2x + 5 ≥ 1.
- Subtract 5: −2x ≥ −4.
- Divide by −2 and reverse the comparison: x ≤ 2.
- The endpoint 2 is included; shade to the left.
x ≤ 2, or (−∞, 2]. The graph has a filled endpoint.
Precision: The boundary is shown to at most 8 significant digits; the calculation itself is not rounded.
Load this example in the calculatorTry a variation
Change only a from −2 to 2, so the inequality is 2x + 5 ≥ 1. What is the solution, and which way does the shading point? Predict the answer before you reveal it.
Predict, then reveal
x ≥ −2, or [−2, ∞). Subtracting 5 gives 2x ≥ −4, and dividing by a positive 2 keeps the direction, so the ray is shaded to the right of a filled endpoint at −2.
| a, with b = 5, c = 1 and ≥ | Solution |
|---|---|
| −2 | x ≤ 2 |
| 2 | x ≥ −2 |
| 0 | All real numbers |
Check the assumptions
- This tool solves a single linear inequality, not nonlinear or two-variable systems.
- If a = 0, compare constants directly to obtain all real numbers or no solution.
Common mistake: Keeping the original direction after dividing by a negative coefficient.
Method reference: OpenStax College Algebra: Linear Inequalities. These examples illustrate the method; they do not validate every possible input.
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Graphing Inequalities Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.
Review your inputs
Double-check that all values are correct and match the units or format shown 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.
Read the results
The Graphing Inequalities Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.
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.
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
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.
About This Calculator
Solve ax+b compared with c and display the one-variable solution on a number line. The tool handles negative or zero coefficients and open or included endpoints.
The Theory Behind It
This tool solves one-variable linear inequalities of the form ax+b<c, ax+b≤c, ax+b>c or ax+b≥c. Subtract b, then divide by a. Dividing by a negative number reverses the inequality because it reverses the order of real numbers. The boundary is (c−b)/a; a strict comparison excludes it and a non-strict comparison includes it. The number line uses an open or filled endpoint and shades the satisfying direction. When a is zero, compare b with c directly: every real x satisfies the statement or no real x does. Do not divide by zero or invent a boundary. This is not a solver for quadratic, rational, compound or two-variable inequalities. Substituting a value on each side of the boundary is a useful independent check of the displayed region. This tool solves a single linear inequality, not nonlinear or two-variable systems. If a = 0, compare constants directly to obtain all real numbers or no solution.
Real-World Applications
- •One-variable linear inequalities ax + b compared with c, drawn on a number line.
Frequently Asked Questions
When must the inequality reverse?
Reverse it when dividing both sides by a negative coefficient. Subtracting the same number from both sides does not reverse it.
What does the endpoint style mean?
A filled endpoint includes the boundary for ≤ or ≥. An open endpoint excludes it for < or >.
Worked Examples
Example 1: Work through the linked example
Solve −2x + 5 ≥ 1.
x ≤ 2, or (−∞, 2]. The graph has a filled endpoint.
Common Mistakes & Tips
- !Keeping the original direction after dividing by a negative coefficient.
- !This tool solves a single linear inequality, not nonlinear or two-variable systems.
- !If a = 0, compare constants directly to obtain all real numbers or no solution.
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