Lesson 2: Inequalities

Property

For any two numbers $a$ and $b$:

• $a < b$ means “$a$ is less than $b$.”
• $a \leq b$ means “$a$ is less than or equal to $b$.”
• $a > b$ means “$a$ is greater than $b$.”
• $a \geq b$ means “$a$ is greater than or equal to $b$.”

On the real line, $a < b$ means that $a$ lies to the left of $b$, and $a > b$ means that $a$ lies to the right of $b$.

Examples

• The statement $x > 5$ means $x$ can be any number to the right of 5, like 6, 8.5, or 20.
• The inequality $-3 \leq y$ means $y$ can be $-3$ or any number to its right, like $-2, 0,$ or 10.
• Comparing fractions: $\frac{1}{3} < \frac{1}{2}$ is true because $\frac{1}{3}$ is to the left of $\frac{1}{2}$ on the number line.

Explanation

Unlike an equation with one answer, an inequality describes a whole range of possible values.
The symbols show whether one number is to the left ($<$) or right ($>$) of another on the number line.

Property

To write an inequality from a verbal phrase, you match keywords to their corresponding inequality symbols.

• Less than ($<$): "is less than", "is smaller than"
• Greater than ($>$): "is greater than", "is more than"
• Less than or equal to ($\leq$): "is at most", "is no more than", "maximum"
• Greater than or equal to ($\geq$): "is at least", "is no less than", "minimum"