Inequality Calculator
Solve linear, quadratic, absolute value, and compound inequalities with steps, a number line, and interval notation.
Solution of 2x + 3 > 7
- Interval notation
- (2, ∞)
- Set-builder notation
- {x ∈ ℝ | x > 2}
- Inequality type
- Linear inequality
- Status
- One interval
- Direction
- Greater than
| Boundary | Endpoint |
|---|---|
| 2 | Excluded (open circle) |
Number line
closed circle = boundary included (≤, ≥) · open circle = boundary excluded (<, >) · arrow = continues to infinity
Step-by-Step Solution
Step 1 — Problem
2x + 3 > 7
Step 2 — Subtract 3 from both sides
2x + 3 − 3 > 7 − 3
2x > 4
Step 3 — Divide both sides by 2
2x ÷ 2 > 4 ÷ 2
x > 2
Step 4 — Solution
x > 2
Interval notation: (2, ∞)
Check a value
Inequality symbols and notation
| Symbol | Meaning | Number line |
|---|---|---|
| < | less than | ○ open, shade left |
| > | greater than | ○ open, shade right |
| ≤ | less than or equal to | ● closed, shade left |
| ≥ | greater than or equal to | ● closed, shade right |
| Inequality | Interval |
|---|---|
| x > 2 | (2, ∞) |
| x ≤ −3 | (−∞, −3] |
| −2 < x ≤ 5 | (−2, 5] |
| x < −3 or x ≥ 2 | (−∞, −3) ∪ [2, ∞) |
| All real numbers | (−∞, ∞) |
| No solution | ∅ |
( ) excludes an endpoint, [ ] includes it, and ∞ always takes a parenthesis because infinity is not a number you can reach.
Solutions are exact: fractions stay fractions and irrational boundaries are shown as square roots, and every answer is re-checked against your inequality before it is displayed. For equations with an equals sign, use the Linear Equation Calculator; for two equations at once, the System of Equations Calculator; for the roots of cubic and higher polynomials, the Polynomial Root Calculator; and for everyday arithmetic, the Scientific Calculator.
This inequality calculator solves inequalities the way you would on paper, then shows the answer in every standard form. Type an inequality such as 2x + 3 > 7, −3x + 5 ≤ 14, x² − 4 < 0, |x − 3| ≤ 2, or −2 < x ≤ 5, and it gives the solution, interval notation, set-builder notation, and each boundary value, marked included or excluded.
The solution is drawn on a number line with open and closed circles, every step is shown (including the step where the sign reverses), and you can check any value of x against the original inequality. It also recognizes when every real number is a solution, or when there is no solution at all.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Basic linear: 2x + 3 > 7 | x > 2, (2, ∞) | Subtract 3 from both sides: 2x > 4. Divide by 2, a positive number, so the sign stays: x > 2. |
| Negative coefficient: −3x + 5 ≤ 14 | x ≥ −3, [−3, ∞) | Subtract 5: −3x ≤ 9. Divide by −3 and reverse the sign: x ≥ −3. |
| Variables on both sides: 3x + 2 > 5x − 4 | x < 3, (−∞, 3) | Subtract 5x: −2x + 2 > −4. Subtract 2: −2x > −6. Divide by −2 and reverse the sign: x < 3. |
| Quadratic: x² − 4 < 0 | −2 < x < 2, (−2, 2) | x² − 4 = (x + 2)(x − 2) is 0 at x = −2 and x = 2. Testing x = 0 gives −4 < 0, so the middle region is the solution; the roots are excluded because < does not allow 0. |
| Compound: −2 < x ≤ 5 | (−2, 5] | Both parts must hold: x > −2 and x ≤ 5. On the number line, −2 has an open circle, 5 has a closed circle, and the segment between them is shaded. |
| Absolute value: |x − 3| ≤ 2 | 1 ≤ x ≤ 5, [1, 5] | Rewrite as −2 ≤ x − 3 ≤ 2, then add 3 to each part: 1 ≤ x ≤ 5. |
What Is an Inequality?
An inequality compares two expressions that are not necessarily equal, using < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). An equation like 2x + 3 = 7 has one answer, x = 2. The inequality 2x + 3 > 7 is true for every x greater than 2, so its solution is a whole range of numbers.
How to Solve an Inequality
Solve a linear inequality like an equation: simplify each side, move the x-terms to one side and the numbers to the other, then divide by the coefficient of x. For 2x + 3 > 7, subtract 3 to get 2x > 4, then divide by 2 to get x > 2.
For a quadratic inequality such as x² − 4 < 0, find where the expression equals 0 (x = −2 and x = 2), then test one number in each region. At x = 0 the expression is −4, which is negative, so the solution is the middle region, −2 < x < 2. For an absolute value inequality, |x − 3| ≤ 2 means x − 3 is within 2 of zero, so −2 ≤ x − 3 ≤ 2, which gives 1 ≤ x ≤ 5.
When Do You Flip the Inequality Sign?
Reverse the sign whenever you multiply or divide both sides by a negative number. In −3x + 5 ≤ 14, subtracting 5 gives −3x ≤ 9. Dividing by −3 reverses ≤ to ≥, so x ≥ −3. You can check it: x = 0 works (5 ≤ 14), but x = −4 does not (17 ≤ 14 is false). Adding, subtracting, and multiplying or dividing by a positive number never change the sign.
How to Graph an Inequality on a Number Line
Mark each boundary with a circle. Use an open circle when the boundary is excluded (< or >) and a closed, filled circle when it is included (≤ or ≥). Then shade the numbers that are solutions: shade to the left for smaller values (x < 2) and to the right for larger values (x > 2). A compound inequality like −2 < x ≤ 5 is shaded between its two boundaries, and x < −3 or x ≥ 2 is shaded outward from both.
Understanding Interval Notation
Interval notation writes a solution as its endpoints. A parenthesis ( ) means the endpoint is excluded and a square bracket [ ] means it is included, so x > 2 is (2, ∞) and x ≤ −3 is (−∞, −3]. Infinity, ∞, always takes a parenthesis because it is not a number that can be reached. Separate pieces are joined with ∪ (union), as in (−∞, −3) ∪ [2, ∞). All real numbers is (−∞, ∞), and no solution is the empty set, ∅.
How to Use the Inequality Calculator
- Type the inequality, such as 2x + 3 > 7, or build it with the keypad. Use <=, >= or the ≤ and ≥ keys, x^2 or x² for squares, |x − 3| for absolute value, a chain like −2 < x ≤ 5, or two parts joined by "or" / "and".
- Press Solve, or tap an example. The result gives the solution, interval notation, set-builder notation, and each boundary value, marked included or excluded.
- Read the number line: a closed circle means the boundary is included (≤ or ≥), an open circle means it is excluded (< or >), and the shaded part is every solution.
- Follow the step-by-step solution. A highlighted step marks where the sign reverses because both sides were multiplied or divided by a negative number.
- Use Check a value to substitute any number and see whether it satisfies the original inequality.
Frequently Asked Questions
What is an inequality?
A statement that two expressions are not equal, written with <, >, ≤, or ≥. Its solution is usually a range of numbers rather than a single value: 2x + 3 > 7 is true for every x > 2.
How do you solve an inequality?
Use the same steps as for an equation: simplify, move x-terms to one side and numbers to the other, then divide by the coefficient of x. Reverse the sign if you multiply or divide by a negative number.
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number. For −3x ≤ 9, dividing by −3 gives x ≥ −3. Swapping the two sides also reverses the symbol: 5 > x means x < 5.
What is interval notation?
A way to write a solution set using its endpoints: parentheses for excluded endpoints and square brackets for included ones. x > 2 is (2, ∞), and −2 < x ≤ 5 is (−2, 5].
What is the difference between an open and a closed circle?
On a number line, an open circle means the boundary is not part of the solution (< or >). A closed, filled circle means it is included (≤ or ≥).
How do you graph an inequality on a number line?
Draw a circle at each boundary, open if excluded and closed if included, then shade the solutions: to the left for "less than", to the right for "greater than", or between the boundaries for a compound inequality.
What does ∞ mean in interval notation?
The solution continues forever in that direction. (2, ∞) means every number greater than 2. Infinity always uses a parenthesis because it is not an actual number that can be included.
What is the difference between an equation and an inequality?
An equation says two expressions are equal and usually has one or a few solutions. An inequality says one is smaller or larger, and usually has a whole range of solutions.
Can an inequality have more than one solution?
Usually it has infinitely many. x > 2 includes 2.5, 3, 100, and every other number above 2. A quadratic or absolute value inequality can even have two separate ranges, such as x ≤ −3 or x ≥ 3.
What does an empty set mean?
No number satisfies the inequality, written ∅. For example, x² + 1 < 0 has no solution because x² + 1 is always at least 1, and |x − 1| < −2 has none because an absolute value is never negative.
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Last updated: September 27, 2026.