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Solving Linear Inequalities 6.1-6.3 Verbal Phrase Inequality Interval Notation 35 Graph All real numbers less than 2 All real numbers greater than −2 All real numbers less than or equal to 1 All real numbers greater than or equal to 0 When you just want greater than or less than, you use an on the graph. When you want (greater than or equal to) or (less than or equal to), you’ll use Try these: 1. x > 3 2. x < −1 3. x ≥ 2 4. x ≤ 0 Graph Interval Notation What if an equation is not quite ready to graph? You already know how to get a variable by itself . . . that is just what you’ll do here!! Example: x + 3 < 10 −3 −3 x < 7 Undo the added 3 by subtracting 3 from both sides Steps: Isolate the variable by using the opposite operation Be sure the variable is on the LEFT hand side of the inequality to graph correctly < > use OPEN circles use CLOSED circles Shade in the direction of the “arrow” Try these: Inequality 1. x 7 12 Solution 2. 9 x 3 3. 4 x 5 4. x 8 12 Interval Not. Graph 5x > 20 Example: 5x 20 Undo the multiplied 5 by dividing both sides by 5 5 5 x > 4 The only time this gets difficult is if you have to divide both sides or multiply both sides by a NEGATIVE number!!! So, when you multiply or divide both sides by a negative number, just flip the inequality!! Example −3x > 15 3x 3 > 15 3 As soon as I divide by the negative number, I circle the inequality to remind me to change it on the next line!! x < − 5 Example x 2 4 4 x 1 4 x > 8 < −2(−4) Try These: Inequality 1. 18 9x 2. x 12 4 3. 15x 30 4. 20 Solution Interval Not. Graph 5 x 4 Multi-Step Inequalities Solving multi-step inequalities is very similar to solving equations with the exception of when to flip the symbol. Remember to flip the symbol when you ____________________! Steps to solving inequalities: Isolate the variable to the LEFT side of the inequality Add or subtract to isolate the variable term Multiply or divide to isolate the variable Remember to FLIP the symbol when necessary Try These: 1. 2x 14 4x 4 2. 6x 3 3x 2 4. 2 6x 9 6x 18 3 5. 2 x 8 4 3 3. 2x 4 6x 4 6. 8 23x 4 10x 20 Class Work: Write your answer in interval notation and graph the answers. 1. x 1 7 2. 42 6x 3. 7 3x 16 4. 2x 10 x 1 5. x 3 7 6. 2x 3 4x 7 7. 17 x 11 4x 31 8. 3(2 x 9) 4( x 3) 1 2 6.1-6.3 Solving Inequalities Homework Solve each inequality. Write your answer in interval notation and graph the answers. 1. 2m + 7 > 17 2. -2 – 3x ≥ 2 3. 7x - 1 < 26 - 2x 4. 2x + 5 < 3x - 7 5. 2x 3 7 5 6. 9r + 15 ≥ 24 + 10r 7. 4y + 2 < 8y – (6y – 10) 8. 5(2h – 6) – 7(h + 7) > 4h 36 9. 3x −5>15−2x 10. 2x−3(1−2x)≤31 11. x ≥1− x 12. 2x <3−4x 13. x ≥2+3x 14. −3x <2 15. 2.3 ≥ −0.2x 16. −2x <1− x