A linear inequality in one unknown compares a linear expression with a value or another linear expression using symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). Instead of finding one exact value, we usually find a set of values that make the statement true.
| Symbol | Meaning | Example |
|---|---|---|
| \(<\) | Strictly less than | \(x < 4\) |
| \(>\) | Strictly greater than | \(x > -2\) |
| \(\leq\) | Less than or equal to | \(x \leq 7\) |
| \(\geq\) | Greater than or equal to | \(x \geq 0\) |
Solve them much like linear equations: expand brackets when needed, collect like terms, then isolate the variable.
Example. Solve \(3x - 5 \leq 16\).
Add \(5\) to both sides: \(3x \leq 21\). Divide by \(3\): \(x \leq 7\).
When you multiply or divide both sides of an inequality by a negative number, the inequality sign must reverse.
Key rule
\[ a < b \;\Rightarrow\; -a > -b \]
Example. Solve \(-2x + 1 > 9\).
Subtract \(1\): \(-2x > 8\). Divide by \(-2\), so reverse \(>\) to \(<\): \(x < -4\).
The solution set can be written on a number line or in interval notation:
| Inequality | Interval notation | Endpoint style |
|---|---|---|
| \(x < 3\) | \((-\infty, 3)\) | Open at \(3\) |
| \(x \leq 3\) | \((-\infty, 3]\) | Closed at \(3\) |
| \(x > -1\) | \((-1, \infty)\) | Open at \(-1\) |
| \(x \geq -1\) | \([-1, \infty)\) | Closed at \(-1\) |
Compound inequalities combine two inequalities with and or or.
Example (and). Solve \(2 < x + 1 \leq 6\).
Subtract \(1\) from all three parts: \(1 < x \leq 5\). Interval form: \((1, 5]\).
Example (or). Solve \(x - 4 < -7\) or \(3x + 1 \geq 10\).
First: \(x < -3\). Second: \(3x \geq 9\), so \(x \geq 3\). Final set: \((-\infty, -3) \cup [3, \infty)\).
Translate the condition into an inequality, solve, and interpret the answer in context.
Example. A bus ticket costs \$12. You have at most \$90. How many tickets \(n\) can you buy?
\(12n \leq 90 \Rightarrow n \leq 7.5\). Since \(n\) is a whole number of tickets, the maximum is \(7\).