Section 1.7: Solving Inequalities
Solving inequalities is similar to solving equations, but the solution is often a set of values. The steps involve isolating the variable, applying the rules of arithmetic, and remembering to reverse the inequality sign when multiplying or dividing by a negative number.
Solutions are written as intervals, on number lines, or in set-builder notation.
- \( x < 4 \) → interval: \( (-\infty, 4) \)
- \( x \geq -2 \) → interval: \( [-2, \infty) \)
Example 1
Solve: \( 3x + 2 \leq 11 \).
Subtract 2: \( 3x \leq 9 \)
Divide by 3: \( x \leq 3 \)
Solution: All \( x \) less than or equal to 3.
Example 2
Solve: \( -5x > 10 \).
Divide by -5 (reverse inequality):
\( x < -2 \)
Solution: All \( x \) less than -2.
Example 3
Solve and represent the solution on a number line: \( 2 \leq x + 1 < 5 \).
Subtract 1 across all parts:
\( 1 \leq x < 4 \)
Solution: \( x \) is between 1 and 4, including 1 but not 4.
Practice Problems
- Solve: \( 4x - 7 < 9 \).
- Solve: \( -2x + 3 \geq 11 \).
- Solve: \( \frac{x}{4} - 2 \leq 1 \).
- Solve: \( 5 \leq 2x + 1 < 11 \).
- Graph the solution set of \( x \geq -1 \) on a number line.