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Inequality Solver

Solve one-variable mathematical inequalities online. Explore solution intervals, approximate roots, and sign charts for polynomial and other arithmetic expressions. Enter both sides of your inequality to get started.

Enter your inequality

Use x as the variable. Write multiplication explicitly, for example 3*x.

Roots outside this range will not be found. A larger range may take longer.

How to solve an inequality

  1. Put both expressions into the calculator. Enter the left side and the right side separately.
  2. Choose the inequality symbol. Select less than, less than or equal to, greater than, or greater than or equal to.
  3. Find critical values. Roots are points where the difference between the two sides is zero. They can divide the number line into intervals.
  4. Check the sign in each interval. Test values help determine where the inequality is true.
  5. Write the solution in interval notation. Use unions when the solution consists of multiple intervals.

This is the standard sign-analysis approach for polynomial inequalities. For rational inequalities, values that make a denominator zero must also be excluded from the domain. See the polynomial and rational inequalities guide.

Example: solve a quadratic inequality

Consider the inequality:

x² − 5x + 6 > 0

First, factor the quadratic expression:

(x − 2)(x − 3) > 0

The roots are \(x = 2\) and \(x = 3\). These divide the number line into three intervals:

  • For \(x < 2\), both factors have the same sign, so the product is positive.
  • For \(2 < x < 3\), the factors have opposite signs, so the product is negative.
  • For \(x > 3\), both factors are positive, so the product is positive.

Because the inequality is strict, the roots are excluded. The exact solution is:

(-∞, 2) ∪ (3, ∞)

You can enter this example above to compare the numerical solver's output with the algebraic solution.

Supported syntax and limitations

The expression parser supports:

  • Variable: x
  • Constants: pi and e
  • Arithmetic: +, -, *, /
  • Exponentiation: ^
  • Parentheses: (x+2)*(x-3)
Important: This page is a numerical, general-purpose exploratory solver. It samples a finite range and looks for sign changes. It can miss repeated roots that do not cross the axis, very close roots, roots outside the selected range, and some discontinuities. The interval output near rational-expression poles may be incomplete. For exact answers, factor the expression or use a symbolic algebra method and verify the result.

Frequently asked questions

What does an inequality solver do?

An inequality solver helps identify the values of a variable that make an inequality true. It can present the answer as intervals and show approximate roots and sign information.

How do I enter an inequality?

Enter the left and right expressions separately, then choose less than, less than or equal to, greater than, or greater than or equal to. For example, enter x^2 - 5*x + 6 on the left, 0 on the right, and choose greater than.

Which mathematical operators are supported?

Use +, -, *, /, ^, parentheses, the variable x, and the constants pi and e. Write multiplication explicitly, such as 2*x rather than 2x.

Are the results exact?

The solver uses numerical sampling and approximate root finding over a finite search range. It is useful for exploration but may miss repeated roots, roots outside the range, or important features near discontinuities. Verify important answers algebraically.

What is interval notation?

Interval notation describes a set of real numbers. Parentheses exclude an endpoint, square brackets include an endpoint, and the union symbol combines separate intervals.