Linear Inequalities: Solving and Graphing

Linear Inequalities are similar to linear equations, but instead of an equals sign (\(=\)), they use inequality symbols (\(<\), \(>\), \(\leq\), \(\geq\)). The solution is usually a range of values rather than a single number.

The Golden Rule of Inequalities

You solve inequalities exactly like equations (add/subtract, multiply/divide), with one major exception:

If you multiply or divide both sides by a negative number, you must FLIP the inequality symbol.

• \( < \) becomes \( > \)
• \( \geq \) becomes \( \leq \)

Step-by-Step Example

Solve and graph:

$$-3x + 5 \geq 14$$

Step 1: Isolate the variable term
Subtract 5 from both sides.

$$-3x \geq 9$$

Step 2: Solve for \(x\)
Divide by \(-3\). Since we are dividing by a negative number, we flip the symbol.

$$x \leq \frac{9}{-3}$$

$$x \leq -3$$

Step 3: Graph on a Number Line

1. Draw a number line.
2. Place a closed circle at \(-3\) (because the symbol is \(\leq\), meaning “included”).
3. Shade the line to the left (values less than \(-3\)).

Solution:
\(x \leq -3\) or \((-\infty, -3]\) in interval notation.

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Why reading isn’t enough

Forgetting to flip the sign is the most common mistake in inequality problems. It’s a small rule with a big impact on your answer.

Weekzen provides targeted practice for linear inequalities. We challenge you with problems that specifically test the negative division rule, ensuring you never miss it on a test again.