Section 1.6: Inequalities Basics
An inequality compares two expressions using symbols such as:
- \( < \): less than
- \( \leq \): less than or equal to
- \( > \): greater than
- \( \geq \): greater than or equal to
Inequalities describe a range of solutions instead of just one solution, unlike equations.
Key Rule:
If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Example 1
Solve: \( x + 3 < 7 \).
Subtract 3 from both sides:
\( x < 4 \).
Solution: All values of \( x \) less than 4.
Example 2
Solve: \( -2x \geq 6 \).
Divide both sides by -2 (reverse inequality):
\( x \leq -3 \).
Solution: All values of \( x \) less than or equal to -3.
Practice Problems
- Solve: \( x - 5 \geq 2 \).
- Solve: \( 3x < 12 \).
- Solve: \( -4x > 20 \).
- Solve: \( \frac{x}{2} + 1 \leq 5 \).
- Solve: \( 2x + 7 \geq 1 \).