Inequality Calculator

About the Inequality Calculator

An equation asks where two expressions are equal; an inequality asks where one is larger or smaller than the other. The answer is usually not a single number but a range of numbers, such as all $x$ between 2 and 3. Writing that range correctly, with the right brackets and endpoints, is where most marks are lost.

This calculator solves linear inequalities like $3x - 5 \ge 7$ , polynomial inequalities like $x^2 - 5x + 6 < 0$ , rational inequalities like $\frac{x+1}{x-2} \le 0$ , compound inequalities like $-3 < 2x + 1 \le 9$ , and absolute value inequalities like $|x - 4| < 3$ . It always shows which points split the number line and tests each interval, so you see why the answer is what it is.

What is an inequality?

An inequality compares two expressions with one of the symbols $<$ , $>$ , $\le$ or $\ge$ . Its solution set is every value of the variable that makes the comparison true. For $x + 2 > 5$ the solution set is every number greater than 3, written $(3, \infty)$ .

Interval notation uses round brackets for endpoints that are excluded and square brackets for endpoints that are included. Infinity always gets a round bracket because it is not a number that can be reached. Separate pieces are joined with the union symbol, as in $(-\infty, 1) \cup [4, \infty)$ .

How to use the Inequality Calculator

Type the inequality with <, >, <= or >=, for example x^2 - 5x + 6 < 0 or 2x + 3 >= 7.
For compound inequalities, write both comparisons in one line, such as -3 < 2x + 1 <= 9.
For absolute values use abs(), for example abs(x - 4) < 3.
Press Solve. Check the critical points and the sign chart, then read the solution set in interval notation.

Formulas

Reversing the sign

a < b \text{ and } k < 0 \Rightarrow ka > kb

Multiplying or dividing by a negative number flips the inequality.

Absolute value, less than

|u| < c \iff -c < u < c \quad (c > 0)

Absolute value, greater than

|u| > c \iff u < -c \ \text{or}\ u > c

Sign of a product

(x - r_1)(x - r_2) < 0 \iff r_1 < x < r_2 \quad (r_1 < r_2)

Worked example: x² − 5x + 6 < 0

In the example, $x^2 - 5x + 6 < 0$ is factored as $(x - 2)(x - 3) < 0$ . The critical points 2 and 3 split the number line into three intervals, and the engine tests one value in each.

Only the middle interval gives a negative product, and the endpoints are excluded because the inequality is strict, so the answer is the open interval $(2, 3)$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

(2, 3)

How we read your problem

Solve the inequality and give the answer in interval notation.

{x}^{2}-5x+6 < 0

Step-by-step solution

Move everything to one side and factor
A sign chart needs the form f(x) compared with 0, written as a product/quotient of factors.
$(x - 3)(x - 2) < 0$
Find the critical points
The sign can only change at 2 (zero), 3 (zero).
Test one point in each interval (sign chart)
(-∞, 2): test x = 1 → sign + → excluded; (2, 3): test x = 2.5 → sign − → included; (3, ∞): test x = 4 → sign + → excluded
Decide the endpoints
2: excluded; 3: excluded
Write the solution set
$x \in (2, 3)$

Formulas used

Reversing the sign
$a < b,\; c < 0 \Rightarrow ac > bc$
Absolute value
$|A| < c \iff -c < A < c,\qquad |A| > c \iff A < -c \text{ or } A > c$

The inequality holds exactly for x in (2, 3). Points where the expression is undefined are never included, and endpoints are included only for ≤ or ≥ where the expression equals zero.

Verification

x = 1: does not satisfy the inequality.
x = 2.5: satisfies the inequality.
x = 4: does not satisfy the inequality.

Detailed explanation

Linear inequalities

Solve a linear inequality exactly like an equation, with one extra rule: whenever you multiply or divide both sides by a negative number, reverse the direction of the sign. From $-2x > 6$ you get $x < -3$ , because multiplying by $-\tfrac12$ reverses the order of numbers on the line.

The reason is that negation mirrors the number line. If $5 > 2$ , then $-5 < -2$ . Adding or subtracting the same number on both sides never changes the direction.

The sign chart method

For polynomial and rational inequalities, move everything to one side so the other side is zero. Then find the critical points: the zeros of the numerator and, for rational expressions, the zeros of the denominator. A polynomial can only change sign at a zero, so between consecutive critical points its sign is constant.

Pick any test value inside each interval and evaluate the sign. Keep the intervals where the sign matches the inequality. Finally decide each endpoint: zeros of the numerator are included for $\le$ or $\ge$ , but zeros of the denominator are always excluded because the expression is undefined there.

Repeated roots deserve attention. In $(x - 1)^2 (x + 2) > 0$ , the factor $(x - 1)^2$ never changes sign, so the expression keeps the same sign on both sides of $x = 1$ . A sign chart handles this automatically, while the shortcut of alternating signs does not.

Compound and absolute value inequalities

A compound inequality like $-3 < 2x + 1 \le 9$ is solved by doing the same operation to all three parts at once, giving $-2 < x \le 4$ . An absolute value inequality is turned into a compound one: $|x - 4| < 3$ means the distance from $x$ to 4 is less than 3, so $1 < x < 7$ .

Common mistakes

Forgetting to flip the sign after multiplying or dividing by a negative number.
Multiplying both sides by an expression like $x - 2$ whose sign is unknown. Use a sign chart instead.
Including a value that makes a denominator zero in the solution set.
Using square brackets next to infinity.
Writing $|x| > 3$ as $-3 > x > 3$ , which no number satisfies; it means $x < -3$ or $x > 3$ .

Tips

Get zero on one side before factoring.
Draw the number line with critical points every time; it prevents most endpoint errors.
Check one value from your answer and one value outside it in the original inequality.
Strict signs give round brackets; inclusive signs give square brackets at numerator zeros.

Frequently asked questions

Why does the inequality sign flip when dividing by a negative number?

Multiplying by a negative number reverses the order of numbers on the number line. Since 5 is greater than 2 but minus 5 is less than minus 2, the comparison must reverse to stay true.

What is interval notation?

A way to write a solution set using brackets. Round brackets exclude an endpoint, square brackets include it, and pieces are joined with a union symbol.

How do I solve a quadratic inequality?

Move everything to one side, factor or find the roots, mark them on a number line, test a value in each interval, and keep the intervals where the sign matches.

Why are some endpoints excluded in rational inequalities?

Values that make the denominator zero make the expression undefined, so they can never be part of the solution, even with less-than-or-equal or greater-than-or-equal.

Can an inequality have no solution?

Yes. For example x squared plus 1 is less than 0 has no real solution because x squared plus 1 is always positive. The calculator reports the empty set.

Can an inequality be true for all real numbers?

Yes. For instance x squared is greater than or equal to 0 holds for every real x, and the calculator answers with the whole real line.

Solve an inequality

Enter your inequality above to see the critical points, the sign chart and the solution in interval notation.

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