What is an inequality in math?

An inequality is a statement that compares two quantities and shows that one is greater than, less than, or possibly equal to the other using symbols like , ≤, or ≥. It usually has many solutions, not just one value.

What is an inequality used for in real life?

Inequalities are used to describe limits, requirements, and ranges. Examples include age limits, spending budgets, speed limits, and score requirements in games, all of which can be written and solved using inequalities.

How do you solve an inequality?

To solve an inequality, treat it much like an equation: add, subtract, multiply, or divide both sides by the same number to isolate the variable. The key difference is that when you multiply or divide both sides by a negative number, you must reverse the inequality sign.

How do you know if a number is a solution to an inequality?

To check whether a number is in the solution set of an inequality, substitute it for the variable and see if the statement is true. If the inequality is true after substitution, the number is a solution; if it is false, the number is not part of the solution set.

What Is an Inequality? Definition and Examples

Kid-friendly explanation

Think of an inequality as a math way to say one side is bigger, smaller, or at least as big as the other side. If you have more game points than a friend, that is like using ">". If you are allowed to have up to 30 minutes of screen time, that is like using "≤ 30". Inequalities help show all the numbers that can work, not just one number.

Step-by-step examples

Example 1 (simpler, one-step inequality)

A snack box can hold at most 12 granola bars. You already put 5 bars in. How many more bars, x, can you add?

Write an inequality: 5 + x ≤ 12.

To solve, subtract 5 from both sides:

5 + x − 5 ≤ 12 − 5, so x ≤ 7.

This means you can add 7 or fewer bars: 0, 1, 2, 3, 4, 5, 6, or 7 bars all make the inequality true.

Example 2 (more advanced, multi-step linear inequality)

Suppose a school club is selling tickets. The club needs to earn more than $250. Each ticket costs $8, and there is a $34 fee to rent the room. How many tickets, t, must they sell so their profit (money after the fee) is greater than $250?

The total money from tickets is 8t. After paying the $34 fee, the profit is 8t − 34. The inequality is:

8t − 34 > 250.

Step 1: Add 34 to both sides to undo subtracting 34:

8t − 34 + 34 > 250 + 34, so 8t > 284.

Step 2: Divide both sides by 8:

8t ÷ 8 > 284 ÷ 8, so t > 35.5.

They cannot sell half a ticket, so they must sell at least 36 tickets. Any whole number of tickets greater than 35.5 (so 36, 37, 38, and so on) makes the inequality true.

How inequalities work and why they matter

An inequality compares two expressions, such as numbers or algebraic expressions with a variable, and tells which side is larger or whether they can be equal. The symbols are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Unlike many equations, which often have one specific solution, an inequality usually has a whole set of solutions. For example, x ≤ 5 has infinitely many solutions, because any number less than or equal to 5 makes it true: 5, 4, 0, −3, 4.9, and so on.

Solving an inequality in one variable is similar to solving an equation. You use the properties of real numbers to add, subtract, multiply, or divide both sides by the same amount. The solution set is all the values of the variable that make the inequality true.

Going deeper with inequalities

As your child moves into middle and high school, inequalities show up in many forms. They will solve and graph linear inequalities in one variable, such as 3x + 2 ≥ 11, and in two variables, such as y < 2x + 3, which is graphed as a half-plane on a coordinate grid.

They will also work with more complex types, including absolute value inequalities (like |x − 4| ≤ 3), quadratic inequalities (like x² − 5x + 6 > 0), radical inequalities (with square roots), and polynomial or rational inequalities (with expressions made from a polynomial or a fraction of polynomials). These often require finding key points where an expression is zero or undefined, then checking which intervals make the inequality true.

Graphing solution sets, using interval notation (for example, x ≥ 2 written as [2, ∞)), and justifying each step when solving are all important skills. Inequalities also connect to geometry through ideas like the triangle inequality theorem, which compares side lengths in a triangle.

Common mistakes with inequalities

Treating an inequality like it has only one answer. A student might say the solution to x ≤ 4 is just x = 4. Correction: x ≤ 4 has infinitely many solutions, including all numbers less than 4 as well as 4 itself.
Forgetting to reverse the inequality symbol when multiplying or dividing by a negative. For example, starting with −2x > 10 and dividing by −2 to get x > −5 is incorrect. The correct step is x < −5, because multiplying or dividing both sides by a negative number reverses the inequality sign.
Graphing the solution set incorrectly on a number line. A student might solve x > 3 correctly but then shade to the left of 3 or use a closed dot at 3. Correction: for x > 3, use an open circle at 3 and shade to the right to show all numbers greater than 3, but not 3 itself.

How inequalities show up in everyday life

Inequalities appear whenever there is a limit, a minimum, or a range. Examples include "You must be at least 13 years old to sign up" (age ≥ 13), "No more than 4 people in the elevator" (people ≤ 4), or "The game score must be greater than 50 to unlock the next level" (score > 50). These real-world rules are all inequalities.

What is an inequality?