1. What an Inequality Means

An inequality compares two expressions that are not necessarily equal. Instead of one answer, an inequality often has many possible answers.

The symbols:

• \(<\): less than
• \(>\): greater than
• \(\le\): less than or equal to, also called at most
• \(\ge\): greater than or equal to, also called at least

Example 1:
\[ x > 5 \]
means \(x\) can be any number greater than 5, such as 6, 10, or 5.1. It cannot be exactly 5.

Example 2:
\[ x \le 5 \]
means \(x\) can be 5 or any number less than 5.

Words matter:

• At least 12 means \(x \ge 12\).
• More than 12 means \(x > 12\).
• At most 12 means \(x \le 12\).
• Less than 12 means \(x < 12\).

Common mistake: treating \(<\) and \(\le\) as the same. The equal bar tells whether the endpoint is included.