4. Inequalities

An inequality compares two sides that are not necessarily equal, using \(<\) (less than), \(>\) (greater than), \(\le\) (at most), or \(\ge\) (at least). Instead of a single answer, the solution is a whole range of numbers.

Great news: you solve inequalities almost exactly like equations — with one special rule.

Worked example. Solve \(2x + 1 > 5\):

• Subtract 1: \(2x > 4\).
• Divide by 2: \(x > 2\).

We graph this on a number line with an open circle at 2 (because 2 itself is not included) and an arrow pointing right toward all larger numbers.

For \(\le\) or \(\ge\), use a closed (filled) circle because the endpoint IS included.

The special rule: when you multiply or divide both sides by a negative number, flip the inequality sign. Solve \(-3x > 12\): divide by \(-3\) and flip: \(x < -4\).

⚠️ Common mistake: forgetting to flip the sign. \(-3x > 12\) gives \(x < -4\), not \(x > -4\).

💡 Tip: test your answer with one number from the range. For \(x > 2\), try \(x = 3\): \(2(3)+1 = 7 > 5\). True — so the direction is right.