Solving Inequalities Lesson
- 1. ≤ , ≥ , <, > Inequalities Chapter 6
- 2. Solving Inequalities To solve an inequality, use inverse operations and solve it just like an equation. Extra Rule: If you multiply or divide both sides by a negative when doing inverse operations, you must reverse the inequality symbol. The graph is an open circle and arrow if only less than or greater than, or a closed circle and arrow if also equal to.
- 3. Example #1 Solve: −2 x − 7 ≤ 5 +7 +7 −2 x ≤ 12 −2 -2 x ≥ −6 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Greater than or equal to means closed circle and arrow.
- 4. Example #2 x Solve: + 1 > −1 −4 −1 −1 x −4 ( ) > (−2 ) ( −4 ) −4 x<8 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Less than means open circle and arrow.
- 5. Example #3 Solve: 4x − 2 > 2x + 6 − 2x − 2x 2x − 2 > 6 +2 +2 2x > 8 x>4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Less than means open circle and arrow.
- 6. Compound Inequalities A compound inequality is two inequalities with the word “and” or “or” between them. “And” statements mean that both statements must be true and the graph is the intersection of the two answers. “Or” statements mean that only one has to be true and the graph is the union of the two answers.
- 7. Steps for Solving Compound Inequalities Step 1: Separate the problem as two inequalities. Step 2: Solve for each problem. Step 3: Graph each answer on a single number line. The combined graph and inequality is your answer.
- 8. Compound Example of “And” Solve: 3 ≤ 2 x − 3 < 15 3 ≤ 2x − 3 2 x − 3 < 15 6 ≤ 2x 2 x < 18 3≤ x x<9 3≤ x <9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 The answer is the intersection between the two points.
- 9. Compound Example of “Or” Solve: h − 10 < −14 or h + 3 ≥ 2 h − 10 < −14 h+3≥ 2 h < −4 h ≥ −1 h < −4 or h ≥ −1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 The answer is the union of the two answers. It can be either but not both at the same time.
- 10. Compound Example of “Or” Solve: h − 10 < −14 or h + 3 ≥ 2 h − 10 < −14 h+3≥ 2 h < −4 h ≥ −1 h < −4 or h ≥ −1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 The answer is the union of the two answers. It can be either but not both at the same time.