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What Is an Inequality and How to Show It on a Number Line?

In math, an inequality shows the relationship between two values in an algebraic expression that are not equal. Inequality signs can indicate that one variable of the two sides of the inequality is greater than, greater than or equal to, less than, or less than or equal to another value.

Whether a sign is greater than or less than depends on the direction of the inequality sign. If the open part of the sign is turned towards the left side, >, the value on the left side of the sign is greater than the right. If it's turned towards the right side, <, the value on the left is less than the value on the right.

Some inequality symbols will have a line underneath them: ≥ and ≤. This means that the two sides of an inequality expression could potentially be equal. However, not enough is known to prove this.

what is an inequality: Mathematical expression

As you can see in the mathematical expression above, x, is greater than or equal to 7. Because x can be many values, saying it’s equal to 7 wouldn't be a true statement. That's why we must use what is an inequality: Greater than or equal to symbol.

Inequalities on a Number Line

Whenever a linear inequality has a variable and a real number, you can express it on a number line. Here's how to use number lines to show x is greater than positive number 3 and less than or equal to negative number -1:

what is an inequality: Mathematical expression

what is an inequality: Linear inequality on a number line

Mathematical expression

Linear inequality on a number line

Any number line showing a linear inequality must have an open circle for < and > and a closed circle for ≤ and ≥.

Using Interval Notation

When we know an inequality is between two numbers, you can write it in interval notation. Interval notation expresses the location range of an inequality by using brackets for ≥ and ≤ signs and parentheses for < and > signs.

Here's how you would show that y is less than or equal to -4 and 2 is greater than y:

Mathematical expression showing inequality between two numbers

Interval notation expressed on a number line

As you can see, we use an open circle to show that y is less than 2 and a closed circle to show that y is equal to or greater than -4. In linear notation, this is written as:

Linear Notation: [-4, 2)

Answering ‘What Is an Inequality?’

Understanding the concept of inequalities allows us to better understand linear equations and the number line. When we know that a variable, like x or y, is within a specific range of values, we can represent it by shading that range of numbers on the number line.

On this line, we use an open circle for greater than or less than values and a closed circle for equal to or less than and equal to or greater than values. If we know the inequality is between two numbers, we can use brackets and parentheses to show the possible range of the inequality’s values in linear notation.

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