Inequality Calculator

An inequality calculator finds the full range of values that make an inequality true, rather than the single exact number an equation usually gives. Take x + 7 > 9 as a quick example. Solving it isolates x on one side and leaves x > 2, meaning every number greater than 2 works, not just one.

What Types of Inequalities Can an Inequality Calculator Solve?

An inequality calculator is built to handle several distinct types, and each one behaves a little differently once you start solving it.

● Linear inequalities, which involve a variable raised to the first power and solve much like a linear equation.

● Compound inequalities, which combine two conditions with the word "and" or the word "or."

● Quadratic inequalities, which involve a squared variable and often produce a solution split across two intervals.

● Rational inequalities, which place a variable inside a denominator and require watching out for values that make that denominator zero.

● Absolute value inequalities, which split into two separate cases once the absolute value bars are removed.

One symbol works differently from the rest. A not-equal-to sign excludes a single point from the number line rather than describing a range, so it is not solved the same way as the four range-based symbols above. Common mistakes and the calculator's limits are covered further down the page.

What Is the Difference Between an Equation and an Inequality?

An equation uses an equals sign and typically has one exact solution, or occasionally a small handful of them. An inequality instead uses greater than, less than, or one of their inclusive versions, and it describes an entire range of values that all satisfy the statement.

Compare x + 2 = 5 with x + 2 > 5. The equation has a single answer, x = 3. The inequality has every value of x greater than 3, an infinite stretch of numbers rather than a single point.

How Do You Solve a Linear Inequality?

Solving a linear inequality follows the same inverse-operation steps used for a linear equation, adding, subtracting, multiplying, or dividing both sides to isolate the variable. There is one exception that matters more than any other rule on this page: multiplying or dividing both sides by a negative number flips the direction of the inequality sign.

Take 2x + 3 > 7 first, a case with no sign flip. Subtracting 3 from both sides gives 2x > 4, and dividing by 2 gives x > 2, with the inequality sign staying exactly as it was.

Now take -3x < 9, a case that requires a flip. Dividing both sides by -3 gives x > -3, not x < -3, because the division was by a negative number. This flip rule is considered the single most common inequality error, so it deserves extra attention every time a negative number shows up on either side of a division or multiplication step.

How Do You Write an Inequality's Solution in Interval Notation?

Interval notation is a shorthand for writing a range of values using brackets and parentheses. A square bracket means the endpoint is included, matching a greater-than-or-equal or less-than-or-equal symbol. A parenthesis means the endpoint is excluded, matching a strict greater-than or less-than symbol.

A solved inequality like x >= 4 converts into the interval [4, infinity). The square bracket next to 4 shows that 4 itself is part of the solution, while the parenthesis next to infinity is just a formatting convention, since infinity is never an endpoint that can be included.

How Do You Graph an Inequality on a Number Line?

A number line graph follows a similar open-versus-closed convention. An open circle marks a strict inequality, greater than or less than only, showing that the endpoint itself is not part of the solution. A closed or filled-in circle marks an inclusive inequality, greater than or equal to or less than or equal to, showing that the endpoint is included. An arrow extending from the circle shows which direction the rest of the solution set stretches.

For x > 2, the graph places an open circle at 2 and draws an arrow extending to the right, since every number greater than 2, but not 2 itself, satisfies the inequality. This connects directly to interval notation above, since both are just different formats for describing the same solution set.

How Do You Solve a Compound Inequality Joined by 'And'?

An "and" compound inequality is true only where both individual inequalities overlap. This overlap is often written as one three-part inequality, such as -1 < x <= 3, meaning x is trapped between two bounds at the same time.

To solve it, isolate the variable in the middle by performing the same operation on all three parts at once. Take -4 < 2x <= 8. Dividing every part by 2 gives -2 < x <= 4. The solution is the intersection of both individual solution sets, and interval notation shows this overlap as one continuous range: (-2, 4].

How Do You Solve a Compound Inequality Joined by 'Or'?

An "or" compound inequality is true wherever either individual inequality is true. Unlike an "and" statement, the two inequalities are solved separately rather than combined into one three-part statement.

Take x < -8 or x > -3 as an example. These two solution pieces point in different directions on the number line, one stretching left from -8 and the other stretching right from -3, with a gap between them where neither condition holds. Since the two pieces do not overlap into one continuous range, interval notation joins them with a union symbol instead: (-infinity, -8) U (-3, infinity).

How Do You Solve an Absolute Value Inequality?

Absolute value inequalities are solved with a two-case split. First isolate the absolute value expression on one side. Then split it into two separate inequalities: one that keeps the original sign with the value on the right, and one that flips the sign and negates that same value.

The pattern that follows is worth memorizing. A less-than-absolute-value inequality turns into an "and" compound inequality, producing one bounded range. A greater-than-absolute-value inequality turns into an "or" compound inequality, producing two separate ranges instead.

For the less-than case, take |x |-2| < 5. This splits into -5 < x - 2 < 5, which solves to -3 < x < 7, a single bounded range. For the greater-than case, take |x |-2| > 5. This splits into x - 2 > 5 or x - 2 < -5, which solves to x > 7 or x < -3, two separate ranges pointing in opposite directions.

What Happens When an Absolute Value Inequality Compares to a Negative Number?

This is a special case worth calling out directly. Since an absolute value can never be negative, an inequality like |x |-3| < -2 has no solution at all, because nothing on the left side can ever be smaller than a negative number. The flip side, |x |-3| > -2, is true for every real number, since the left side is always at least zero, which is already greater than -2. There is no case to split here, so the two-case method covered above simply does not apply.

How Do You Check If an Inequality Calculator's Answer Is Correct?

The most reliable way to check a solution is the test-point method. Pick a number from inside the claimed solution range and substitute it into the original inequality to confirm it holds true. Then pick a number outside the range and confirm the inequality fails for that value.

Take the solved inequality x > 2 as an example. Testing x = 5, a number inside the range, in the original inequality x + 7 > 9 gives 12 > 9, which is true. Testing x = 0, a number outside the range, gives 7 > 9, which is false, exactly as expected. A passing test point inside the range and a failing test point outside it together confirm the solution is correct.

What Are the Limitations of an Inequality Calculator?

An inequality calculator is dependable, but a few details are worth knowing before relying on it fully. It returns a range of values rather than one number, so a reader expecting a single answer the way an equation gives one may find the output unfamiliar at first. A rational inequality, one with a variable in the denominator, needs extra care, since the value that makes that denominator zero must always be excluded from the solution, even if it otherwise seems to fit the pattern. Ambiguous input can also cause trouble, since a missing greater-than-or-equal symbol or unclear grouping around parentheses can produce a wrong or unreadable result even though the underlying math the calculator runs is correct.

How Does Solving One Inequality Connect to Other Algebra Calculators?

Once you are comfortable solving a single inequality, a few related tasks tend to come up naturally. Sometimes the problem actually uses an equals sign instead of an inequality symbol, sometimes the inequality involves two variables and needs to be graphed rather than solved for one, and sometimes an expression needs to be cleaned up before it is even ready to solve. The sections below connect each of those situations back to what was covered above.

How Is the Equation Solver Different From the Inequality Calculator?

The equation solver finds the exact value or values that make both sides equal, while the inequality calculator finds every value that satisfies a greater-than or less-than comparison, which is normally an entire range rather than one number. If a problem uses an equals sign instead of an inequality symbol, the equation solver is the better fit, since the two tools apply mostly the same inverse-operation logic but hand back very different kinds of answers.

How Does a Graphing Calculator Show a Two-Variable Inequality?

A two-variable inequality, like y > 2x + 1, is graphed by first drawing the boundary line, using a dashed line for a strict inequality and a solid line for an inclusive one, then shading the side of the line that satisfies the inequality. A graphing calculator handles that shading automatically and is a quick way to confirm a test-point check done by hand.

How Does a Simplify Calculator Help Before You Solve an Inequality?

Simplifying combines like terms and clears unnecessary parentheses without changing the inequality's meaning, which is a useful first pass before tackling an inequality with terms scattered across both sides. A messy inequality is more likely to produce a sign-flip mistake if it is not cleaned up first. Running it through a simplified calculator before isolating the variable ties directly back to the linear inequality steps covered earlier on this page.

Frequently Asked Questions

How Do You Flip the Inequality Sign When Solving?

Flip the inequality sign only when multiplying or dividing both sides by a negative number, and nowhere else, such as when adding or subtracting a value from both sides.

What Is the Difference Between an 'And' and an 'Or' Compound Inequality?

An "and" compound inequality needs both parts to be true at once, giving one continuous overlapping range. An "or" compound inequality needs only one part to be true, which often gives two separate ranges pointing in different directions.

Can an Absolute Value Inequality Have No Solution?

Yes. An inequality like |x | -3 | < -2 has no solution, since an absolute value can never be less than a negative number. The same setup with a greater-than sign, |x |-3| > -2, is true for every real number instead.

How Do You Write a Solution in Interval Notation?

A square bracket means the endpoint is included in the solution, matching an inclusive symbol like greater-than-or-equal-to, while a parenthesis means the endpoint is excluded, matching a strict symbol like greater-than. For example, x >= 4 becomes [4, infinity).

What Does It Mean if an Inequality Has No Solution or All Real Numbers as Its Solution?

An inequality has no solution when every value fails to satisfy it, such as a false constant statement appearing after the variable cancels out. It has all real numbers as its solution when every value satisfies it, such as a true constant statement appearing instead.