Linear Inequalities
Linear Inequalities
An inequality expresses a range of values rather than a single solution. Solving a linear inequality follows the same steps as solving a linear equation, with one critical exception: multiplying or dividing both sides by a negative number reverses the inequality sign. The solution set is typically represented on a number line.
Inequality Symbols
| Symbol | Meaning | Number line |
|---|---|---|
| \( x > a \) | \(x\) is greater than \(a\) | Open circle at \(a\), arrow right |
| \( x \geq a \) | \(x\) is greater than or equal to \(a\) | Closed circle at \(a\), arrow right |
| \( x < a \) | \(x\) is less than \(a\) | Open circle at \(a\), arrow left |
| \( x \leq a \) | \(x\) is less than or equal to \(a\) | Closed circle at \(a\), arrow left |
The Critical Rule
All other operations (adding, subtracting, multiplying/dividing by positive numbers) leave the sign unchanged.
Double Inequalities
A combined inequality like \( 2 \leq 3x - 1 < 11 \) represents all \(x\) satisfying both parts simultaneously. Solve by performing the same operations across all three parts of the inequality.
\[ 2 \leq 3x-1 < 11 \implies 3 \leq 3x < 12 \implies 1 \leq x < 4 \]Worked Examples
Solve \( 3x - 4 > 11 \) and represent the solution on a number line.
\[ 3x > 15 \implies x > 5 \]Number line: open circle at 5, arrow to the right.
Solve \( 5(2-x) \leq 20 \).
\[ 10-5x \leq 20 \implies -5x \leq 10 \implies x \geq -2 quad \text{(divided by \(-5\), flip sign)} \]Solve \( 7 - 2x < 3 \) and list the positive integers in the solution set.
\[ -2x < -4 \implies x > 2 quad \text{(divided by \(-2\), flip sign)} \]Positive integers greater than 2: 3, 4, 5, 6, …
Solve \( 4 - 3x \geq 13 \).
\[ -3x \geq 9 \implies x \leq -3 \]Solve \( -1 < 2x + 3 \leq 11 \) and represent on a number line.
\[ -4 < 2x \leq 8 \implies -2 < x \leq 4 \]Number line: open circle at \(-2\), closed circle at 4, solid line between them.
Find the integers satisfying \( 3 < 4x - 1 \leq 15 \).
\[ 4 < 4x \leq 16 \implies 1 < x \leq 4 \]Integers: 2, 3, 4.
Key Takeaways
- Solve linear inequalities like equations, with the critical exception: multiplying or dividing by a negative flips the sign.
- Open circle on a number line: endpoint excluded (\(>\) or \(<\)). Closed (filled) circle: endpoint included (\(\geq\) or \(\leq\)).
- Double inequality: operate on all three parts simultaneously.
- If the \(x\) term ends up negative, avoid dividing by a negative: alternatively, collect \(x\) terms on the other side first.
- The solution to an inequality is a set of values (often a range), not a single value.