Inequality Calculator

1. Pick the type: a linear inequality such as 5x - 2 >= 3x + 6, or a compound one such as -2 < 3x + 1 <= 10.
2. Type the numbers for each part. Use negative numbers for minus signs, so 5x - 2 has a = 5 and b = -2. Use 0 when a side has no x term.
3. Choose the sign. The answer, the number line, and the interval notation update as you type.
4. Read the steps to see each move, including any sign flip, then press Copy report or Save.

The test value card plugs a number from the answer back into your original inequality, so you can see that it really works.

An inequality compares two amounts that may not be equal. Four signs cover almost every school problem.

Textbooks print the "or equal to" signs with a single bar under the angle. We type them as <= and >= so they work on any keyboard.

You solve a linear inequality with the same moves as an equation. Collect the x terms on one side, move the plain numbers to the other, and then divide.

Worked Example: -3x + 5 <= 14

1. Subtract 5 from both sides: -3x <= 9. Adding or subtracting never changes the sign.
2. Divide both sides by -3. The divisor is negative, so <= turns into >=.
3. The answer is x >= -3. A quick check with x = 0 gives 5 <= 14, which is true.

Why a Negative Number Flips the Sign

Multiplying by a negative number mirrors every value across zero. Numbers that were far to the right end up far to the left, so their order reverses. Start with 2 < 3 and multiply both sides by -1.

On a number line, -2 sits to the right of -3, so -2 is the bigger number. Keeping the old sign would claim -2 < -3, which is false. Division by a negative number works the same way, since it is multiplication by a negative fraction.

Most answers are a whole range of numbers, not a single value. Interval notation and number lines are two short ways to write that range.

• Round bracket ( ): the endpoint is not included. It matches < and >, and an open circle.
• Square bracket [ ]: the endpoint is included. It matches <= and >=, and a closed circle.
• Infinity: written inf here and as the sideways eight in print. It always takes a round bracket, because no number reaches it.

The number line below shows x >= -3. The filled circle means -3 itself works, and the shading runs right toward bigger numbers.

Two special answers need no circle at all. "All real numbers" is the whole line, written (-inf, inf). "No solution" is the empty set, written { }.

A compound, or "between", inequality keeps an expression inside two limits. You solve it by doing the same step to all three parts at once.

The answer is -1 < x <= 3, or (-1, 3]. It has an open circle at -1, a closed circle at 3, and shading between them.

With a negative coefficient, both signs flip. Then rewrite the result so the smaller number sits on the left. For 4 <= -2x <= 10, dividing by -2 gives -2 >= x >= -5, which reads as -5 <= x <= -2, or [-5, -2].

Some compound inequalities have no answer. In 5 < x < 3, no number is both above 5 and below 3, so the solution set is empty.

1. Forgetting the Flip

Dividing -3x <= 9 by -3 and writing x <= -3 is the most common error. Test a value to catch it. Try x = -4: then -3(-4) = 12, and 12 <= 9 is false.

2. Flipping When You Should Not

Only multiplying or dividing by a negative flips the sign. Subtracting 5 or adding -5 leaves it alone. So does dividing by a positive number when the answer turns out negative.

3. Mixing Up Circles and Brackets

An open circle goes with a round bracket, and a closed circle goes with a square one. Infinity is never closed, so [3, inf] is wrong. Write [3, inf) instead.

4. Writing a Compound Answer Backwards

After a flip, -2 >= x >= -5 is correct but hard to read. Rewrite it as -5 <= x <= -2, with the smaller number first, before you draw it.

Solve it the same way you would solve an equation. Move the x terms to one side and the numbers to the other, then divide by the x coefficient. The only extra rule is that multiplying or dividing by a negative number flips the sign.

Flip it only when you multiply or divide both sides by a negative number. Adding or subtracting any number, even a negative one, never flips the sign. For example, -3x <= 9 becomes x >= -3 after dividing by -3.

An open circle means the endpoint is not part of the answer, which goes with < or >. A closed, filled circle means the endpoint is included, which goes with <= or >=.

Write the smallest and largest values in brackets. Use a round bracket for an endpoint that is not included and a square bracket for one that is. Infinity always gets a round bracket, so x >= -3 is [-3, inf).

It happens when the x terms cancel. If what is left is true, such as 1 < 2, every x works and the answer is all real numbers. If it is false, such as 2 < 1, no x works and there is no solution.

Do the same step to all three parts at once. Subtract 1 to get -3 < 3x <= 9, then divide by 3 to get -1 < x <= 3. In interval notation that is (-1, 3].

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• OpenStax. Elementary Algebra 2e, section 2.7: Solve Linear Inequalities (number lines, interval notation, the flip rule).
• OpenStax. Intermediate Algebra 2e, section 2.5: Solve Linear Inequalities (properties of inequality, all reals and no solution cases).
• OpenStax. College Algebra 2e, section 2.7: Linear Inequalities and Absolute Value Inequalities (compound inequalities).
• Dawkins, P. Linear Inequalities. Paul's Online Math Notes, Lamar University (worked double inequalities).
• Weisstein, E. W. Inequality. MathWorld, Wolfram Research (definition and notation).