Inequality Solver
Use this inequality solver to solve a one-variable linear inequality in the form ax + b compared with c, with steps, interval notation, and a simple number line.
Algebra calculatorQuick Result Explanation
The result tells you which values of x make the inequality true. If the answer is x <= 4, every number less than or equal to 4 works. If the answer is x > -3, every number greater than -3 works.
For strict signs such as less than or greater than, the boundary number is not included. For less than or equal to or greater than or equal to, the boundary number is included.
Formula and Method
This calculator solves linear inequalities written as:
ax + b compared with c
The method is similar to solving a linear equation:
- Move the constant b to the right side: ax compared with c – b.
- Divide both sides by a.
- If a is negative, reverse the inequality sign.
- Write the answer as a solution statement and interval notation.
How to Use the Inequality Solver
- Enter the coefficient of x in the first box.
- Enter the constant term that is added to ax.
- Choose the inequality sign that matches your problem.
- Enter the right side of the inequality.
- Click Solve Inequality to see the result, interval notation, and steps.
- Use Copy Result if you want to save the visible answer.
Worked Example
Suppose the inequality is 2x + 3 <= 11.
| Step | Work | Reason |
|---|---|---|
| Start | 2x + 3 <= 11 | Original inequality |
| Subtract 3 | 2x <= 8 | Move the constant to the right side |
| Divide by 2 | x <= 4 | 2 is positive, so the sign stays the same |
| Solution | x <= 4 | All numbers up to and including 4 work |
Common Use Cases
Algebra homework
Check one-variable inequality problems and compare your own steps with the calculator output.
Test preparation
Practice how signs change when the coefficient of x is negative.
Word problems
Model limits such as maximum cost, minimum score, or allowed range using a simple inequality.
Quick checking
Verify whether the solution should point left or right on a number line.
Common Mistakes
- Not reversing the sign: When you divide by a negative coefficient, less than becomes greater than, and greater than becomes less than.
- Mixing up strict and inclusive signs: Less than is not the same as less than or equal to.
- Entering the wrong constant: For 3x – 7 < 8, enter b as -7, not 7.
- Expecting quadratic results: This tool solves linear inequalities only, not inequalities such as x squared – 4 > 0.
- Ignoring the no x case: If a is zero, the inequality may be always true or never true.
Limitations
This inequality solver handles one-variable linear inequalities in the form ax + b compared with c. It does not solve compound inequalities, absolute value inequalities, rational inequalities, or quadratic inequalities.
Decimal output may be rounded for readability when the exact value has many digits. For schoolwork, always check whether your teacher expects fractions, decimals, interval notation, or a number line.
Inequality Solver FAQ
What type of inequalities can this calculator solve?
It solves linear inequalities with one variable, such as 2x + 3 <= 11 or -4x + 6 > 18.
When do I reverse the inequality sign?
Reverse the sign when you multiply or divide both sides by a negative number. For example, dividing by -4 changes greater than into less than.
What does interval notation mean?
Interval notation describes the same solution set in a compact form. For example, x <= 4 becomes (-infinity, 4].
Why does the calculator sometimes say all real numbers?
If the x coefficient is zero and the remaining statement is true, then every value of x works because x no longer affects the inequality.
Can this solve quadratic inequalities?
No. Quadratic inequalities need a different method because the solution can contain two intervals or depend on roots of a parabola.
Final Note
This tool is best for checking linear inequality answers quickly while still seeing the algebra behind the result.
