College Algebra - Week 3
Inequalities
An inequality describes a range of values rather than a single answer. This week you learn to solve, graph, and express the solution sets of linear inequalities, compound inequalities, and absolute value inequalities. Interval notation is introduced here and used throughout the rest of the course.
Learning Objectives
- Solve linear inequalities in one variable and graph solutions on a number line. - Express inequality solutions using interval notation. - Understand why the inequality sign flips when multiplying or dividing by a negative. - Solve compound inequalities joined by 'and' (intersection) and 'or' (union). - Solve absolute value equations. - Solve absolute value inequalities of the form |ax + b| < c and |ax + b| > c. - Translate word problems into inequalities and solve.
1. Inequalities vs. Equations
An equation has exactly one solution (usually). An inequality has infinitely many solutions — a whole interval of values. 3x = 12 has one solution: x = 4. 3x < 12 has infinitely many solutions: all x less than 4. The solution to an inequality is a set of numbers, not a single number. We express this set as a graph on a number line or in interval notation.
2. Solving Linear Inequalities
Solve inequalities exactly like equations, with ONE critical rule: When you multiply or divide both sides by a NEGATIVE number, the inequality symbol flips direction. Example 1: Solve 3x + 4 > 13. Subtract 4: 3x > 9. Divide by 3 (positive): x > 3. Example 2: Solve -2x + 5 < 11. Subtract 5: -2x < 6. Divide by -2 (FLIP the sign): x > -3. Graph: open circle at -3, arrow pointing right.
3. Number Line Graphs and Interval Notation
Number line conventions: - Open circle (or parenthesis) at an endpoint means the value is NOT included (strict inequality: < or >). - Closed circle (or bracket) means the value IS included (non-strict: <= or >=). Interval notation: - x > 3: (3, infinity) - x >= 3: [3, infinity) - x < -1: (-infinity, -1) - x <= 5: (-infinity, 5] - -2 < x <= 4: (-2, 4] Always write smaller number on the left. Infinity always uses a parenthesis, never a bracket.
4. Compound Inequalities: AND (Intersection)
An 'and' compound inequality requires BOTH conditions to be satisfied. The solution is the INTERSECTION of both solution sets. Example: Solve -1 < 2x + 3 < 11. Subtract 3 from all parts: -4 < 2x < 8. Divide all parts by 2: -2 < x < 4. Interval notation: (-2, 4). You can also solve each part separately and find what they share.
5. Compound Inequalities: OR (Union)
An 'or' compound inequality requires AT LEAST ONE condition to be satisfied. The solution is the UNION of both solution sets. Example: Solve 2x - 1 < 5 OR 3x + 2 > 14. Left part: 2x < 6 → x < 3. Right part: 3x > 12 → x > 4. Solution: x < 3 OR x > 4. Interval notation: (-infinity, 3) U (4, infinity). These are two separate rays. They do not overlap.
6. Absolute Value Equations
The absolute value |x| measures the distance from x to 0. Distance is never negative. |x| = 5 means x = 5 or x = -5. General rule: |expression| = c (where c > 0) splits into two equations: expression = c OR expression = -c Example: Solve |2x - 3| = 7. Case 1: 2x - 3 = 7 → 2x = 10 → x = 5. Case 2: 2x - 3 = -7 → 2x = -4 → x = -2. Check: |2(5)-3| = |7| = 7. |2(-2)-3| = |-7| = 7. Both correct.
7. Absolute Value: Less Than
Think of |x| < 5 as: distance from 0 is less than 5. Solution: -5 < x < 5 (an AND compound inequality). General rule: |expression| < c becomes -c < expression < c. Example: Solve |3x - 1| < 8. -8 < 3x - 1 < 8 Add 1: -7 < 3x < 9 Divide by 3: -7/3 < x < 3 Interval notation: (-7/3, 3).
8. Absolute Value: Greater Than
Think of |x| > 5 as: distance from 0 is greater than 5. Solution: x < -5 OR x > 5 (an OR compound inequality). General rule: |expression| > c becomes expression < -c OR expression > c. Example: Solve |2x + 1| > 5. Case 1: 2x + 1 < -5 → 2x < -6 → x < -3. Case 2: 2x + 1 > 5 → 2x > 4 → x > 2. Interval notation: (-infinity, -3) U (2, infinity).
9. Special Absolute Value Cases
If |expression| = 0: the expression equals 0. Exactly one solution. If |expression| < 0: impossible. No solution (absolute value is never negative). If |expression| > 0: all real numbers except where the expression equals 0. Example: |x - 3| < 0 → No solution. |x - 3| >= 0 → All real numbers (always true). Always check: if c < 0 on the right side of |...| = c or |...| < c, the answer is no solution without further work.
10. Inequality Word Problems
Example 1: A student needs at least 90 on the final to earn an A. The final is worth 25% of the grade. The student's current average is 87. What score does the student need? 0.75(87) + 0.25x >= 90 65.25 + 0.25x >= 90 0.25x >= 24.75 x >= 99 The student needs at least 99 on the final. Example 2: A temperature must stay between 60 and 80 degrees Fahrenheit. Convert to Celsius: C = (5/9)(F - 32). 60 <= F <= 80 Subtract 32: 28 <= F - 32 <= 48 Multiply by 5/9: 155/9 <= C <= 80/3 (approximately 17.2 to 26.7 Celsius).
11. Common Mistakes
- Forgetting to flip the inequality when multiplying or dividing by a negative. - Using brackets next to infinity symbols: write (-inf, 5], not [-inf, 5]. - For OR inequalities: writing the solution as one interval when it is actually two separate rays. - For |expression| > c: writing one single middle piece instead of two outside rays. - Confusing which case applies: |...| < c → between (AND), |...| > c → outside (OR).