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Representing Inequalities, Solving Inequalities and Absolute Value Inequalities Mathematical sentences that use any of the following symbols > Greater than < Less than ≤ Less than or equal to 44 ≥ Greater than or equal to Solving Inequalities Done the same way you solve equations. Exception: when you multiply or divide both sides of an inequality by a negative number, you must change the direction of the inequality symbol. Example: Solving Inequalities Using Addition/Subtraction Solve the following inequalities and graph the solution on the number line. a. y + 3 > 5 b. x - 3 < 5 Step 1: Isolate y variable Step 1: Isolate the x variable Subtract 3 from both sides add 3 to both sides y+3>5 x-3<5 - 3 > -3 + 3 < +3 y >2 x < 9 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Example: Solving Inequalities Using Multiplication/Division Solve the following inequalities and graph the solution on the number line. a. 4y > 12 b. –3y > 15 Step 1: Isolate y variable divide both sides by 3 4 y 12 > 4 4 y>3 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Step 1: Isolate the y variable divide both sides by -3 3 y 15 > 3 3 y < -5 (since we divided by negative, ineq switched) –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Solving Inequalities (Multi-Step) Complete the Distributive Property Simplify by adding like terms. Eliminate the variable on 1 side Eliminate constant term on the side with variable Solve for the variable Check solution Remember: addition/subtraction must be done before multiplication/division Note: some inequalities have no solution and others are true for all real numbers. Representing Inequalities, Solving Inequalities and Absolute Value Example: Solving Inequalities (Multi-Step) Solve the following inequalities and graph the solution on the number line. a. 2y + 3 < 9 b. 3y + 2y > 15 Step 1: Opposite of add is subtract So subtract 3 from both sides Step 1: Add like terms So add 3y + 2y Step 2: Perform the necessary operation 2y + 3 < 9 - 3 -3 2y <6 Step 2: Perform the necessary operation 5y > 15 Step 3: Opposite of multiply is divide So, divide by 2 Step 3: Opposite of multiply is divide So, divide by 5 Step 4: Perform the necessary operation 2y 6 < 2 2 y<3 Step 4: Perform the necessary operation 5 y 15 > 5 5 y > 3 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Compound Inequalities 2 inequalities joined by the word “and” or “or” Example: -5≤ x ≤ 7 is the same as x ≥ -5 and x ≤ 7 Example: Solving Compound Inequalities (and/or) Solve the following compound inequalities and graph the solution on the number line. a. –4 < r –5 ≤ -1 b. 4v + 3 < -5 or –2v + 7 < 1 Step 1: Isolate the variable r Add 5 to all sides –4 < r –5 ≤ -1 +5 +5 +5 1<r≤4 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Step 1: Isolate the variable v 4v + 3 < -5 or –2v + 7 < 1 -3 -3 -7 -7 2v 6 4 v 8 or ` 4 4 2 2 v < -2 or v > 3 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Representing Inequalities, Solving Inequalities and Absolute Value Absolute Value Equations & Inequalities Since absolute value represents distance, it can never be negative When solving for |a| = b, 2 solutions a = b and a = -b When solving for |a| < b, solving for –b < a < b When solving for |a| > b, solving for a < -b or a > b Example: Solving Absolute Value Equations Solve the following equations. a. | x | + 5 = 11 Step 1: Isolate absolute value function |x | + 5 – 5 = 11 – 5 Step 2: Simplify |x| = 6 Step 3: Write 2 equations & solve x = 6 or x = -6 b. |2p + 5| =11 |2p + 5| = 11 2p + 5 = 11 or 2p + 5 = -11 -5 = -5 -5 = -5 2p = 6 or 2p = -16 p =3 or p = -8 Example: Solving Absolute Value Inequalities Solve the following inequalities. Check and graph your solution. a. |n -1 | < 5 b. |v -3| ≥ 4 n – 1 < 5 or n –1 > -5 v – 3 ≥ 4 or v – 3 ≤ -4 n –1 + 1 < 5 + 1 or n –1 + 1 > -5 +1 v – 3 + 3 ≥ 4 + 3 or v –3 + 3 ≤ -4 + 3 n<6 or n > -4 v≥7 or v ≤ -1 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Representing Inequalities, Solving Inequalities and Absolute Value 45 Representing Inequalities Practice Inequality 1. 5< x < 8 2. x < -3 3. x>0 Interval Graph Set Notation Rewrite as an inequality and graph: 4. [5, 9) 5. [-3, 10] 6. (-, 5] 7. (-3, ) Rewrite in interval notation, set notation, and graph: 8. -6 < x < 6 9. x > -3 10. x < -2 11. -2 < x < 5 12. x < 4 or x > 6 13. x < -2 or x > 0 Write as an inequality, in interval notation and in set notation: 14. 15. -1 -8 3 16. 17. -4 7 3 18. 18 19. 6 -9 Representing Inequalities, Solving Inequalities and Absolute Value Practice: Solving Inequalities Using Addition/Subtraction and Multiplication/ Division Solve the following inequalities. Graph your solution. 1. x – 3 < 5 2. 12 ≤ x – 5 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 3. n – 7 ≤ -2 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 5. 2 n≤2 3 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 7. x < -1 2 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 4. –4 > b - 1 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 6. 6 ≤ 3 w 5 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 8. –20 > -5c –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Practice: Solving Inequalities (Multi-Step) Solve the following inequalities and graph the solution on the number line. 9. -15c – 28 > 152 10. 4x – x + 8 ≤ 35 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Representing Inequalities, Solving Inequalities and Absolute Value 11. 2x – 3 > 2(x-5) 12. 7x + 6 ≤ 7(x – 4) –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Practice: Solving Compound Inequalities (and/or) Solve the following compound inequalities and graph the solution on the number line. 13. –6 < 3x < 15 14. –3 < 2x – 1 < 7 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 15. 7 < -3n + 1 ≤ 13 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 17. 2d + 5 ≤ -1 or –2d + 5 ≤ 5 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 16. –2x + 7 > 3 or 3x – 4 ≥ 5 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 18. 3x + 2 < -7 or –4x + 5 < 1 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Representing Inequalities, Solving Inequalities and Absolute Value Practice: Solving Absolute Value Equations Solve the following equations. Check your solution. 19. |t| -2 = -1 20. 3|n| = 15 21. 4 = 3|w| - 2 Practice: Solving Absolute Value Inequalities Solve the following inequalities. Graph your solution. 22. |w + 2 | > 5 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 24. 4 3s 7 5 7 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 23. |y – 5| ≤ 2 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 25. 2 j 9 2 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10