1. Solving and Graphing Inequalities (5-1, 5-2) Objective: Solve linear inequalities by using addition and subtraction. Solve linear inequalities by using multiplication and division.

2. Solve Inequalities by Addition Addition Property of Inequalities: –If the same number is added to each side of a true inequality, the resulting inequality is also true. –For all numbers a, b, and c, the following are true. 1.If a > b, then a + c > b + c. 2.If a < b, then a + c < b + c. –This property is also true for ≥ and ≤.

3. Example 1 Solve c – 12 > 65. c – 12 > 65 +12 c > 77 {all numbers greater than 77}

4. Check Your Progress Choose the best answer for the following. –Solve k – 4 < 10. A.k > 14 B.k < 14 C.k < 6 D.k > 6 k – 4 < 10 +4

5. Notation A more concise way of writing a solution set is to use set-builder notation. In set-builder notation, the set is written as {variable | inequality}. {k|k < 14} would be read as “the set of all numbers k such that k is less than 14”.

6. Solve Inequalities by Subtraction Subtraction Property of Inequalities: –If the same number is subtracted from each side of a true inequality, the resulting inequality is also true. –For all numbers a, b, and c, the following are true. 1.If a > b, then a – c > b – c. 2.If a < b, then a – c < b – c. –This property is also true for ≥ and ≤.

7. Example 2 Solve the inequality x + 23 < 14. x + 23 < 14 -23 x < -9 {x|x < -9}

8. Check Your Progress Choose the best answer for the following. –Solve the inequality m – 4 ≥ -8. A.{m|m ≥ 4} B.{m|m ≤ -12} C.{m|m ≥ -4} D.{m|m ≥ -8} m – 4 ≥ -8 +4

9. Graphing The solution set can be graphed on a number line. The graph will consist of an endpoint and shading. The endpoint will be a circle for > and <. The endpoint will be a dot for ≥ and ≤. The shading will be to the right for > and ≥. The shading will be to the left for < and ≤. Always graph an inequality with the variable on the left of the inequality sign.

10. Example 3 Solve 12n – 4 ≤ 13n. Graph the solution set. 12n – 4 ≤ 13n -12n -4 ≤ n {n|n ≥ -4} n ≥ -4

11. Check Your Progress Choose the best answer for the following. –Solve 3p – 6 ≥ 4p. Graph the solution. A.{p|p ≤ -6} B.{p|p ≤ -6} C.{p|p ≥ -6} D.{p|p ≥ -6} 3p – 6 ≥ 4p -3p -6 ≥ p

12. Verbal Problems Verbal problems containing phrases like greater than or less than can be solved by using inequalities. The chart shows some other phrases that indicate inequalities. <>≤≥ less than fewer than greater than more than at most no more than less than or equal to at least no less than greater than or equal to

13. Example 4 Panya wants to buy season passes to two theme parks. If one season pass costs $54.99 and Panya has $100 to spend on both passes, the second season pass must cost no more than what amount? –Let c = cost of season pass –She can spend up to $45.01 on the second season pass. c + 54.99 ≤ 100 -54.99 c ≤ 45.01

14. Check Your Progress Choose the best answer for the following. –Jeremiah is taking two of his friends out for pancakes. If he spends $17.55 on their meals and has $26 to spend in total, Jeremiah’s pancakes must cost no more than what amount? A.$8.15 B.$8.45 C.$9.30 D.$7.85 c + 17.55 ≤ 26 -17.55

15. Solve Inequalities by Multiplication Multiplication Property of Inequalities: –If you multiply each side of an inequality by a positive number, then the inequality remains true. –For any real numbers a and b and any positive number c, if a > b, then ac > bc. And, if a < b, then ac < bc. –If you multiply each side of an inequality by a negative number, the inequality symbol changes direction. –For any real numbers a and b and any negative real number c, if a > b, then ac bc. –This property also hold for inequalities involving ≤ and ≥.

16. Example 5 Mateo walks at a rate of ¾ mile per hour. He knows that it is at least 9 miles to Onyx Lake. How long will it take Mateo to get there? Write and solve an inequality to find the time. h ≥ 12 It will take at least 12 hours.

17. Check Your Progress Choose the best answer for the following. –At Midpark High School, 2 / 3 of the junior class attended the dance. There were at least 200 juniors at the dance. How many students are in the junior class? A.j ≤ 300 B.j ≥ 300 C.j ≥ 200 D.j ≤ 200

18. Example 6 Solve d ≤ -10 {d|d ≤ -10}

19. Check Your Progress Choose the best answer for the following. –Solve - 1 / 3 x > 10. A.x > 10 / 3 B.x > - 10 / 3 C.x < -30 D.x > -30

20. Solve Inequalities by Division Division Property of Inequalities: –If you divide each side of an inequality by a positive number, then the inequality remains true. –For any real numbers a and b and any positive real number c, if a > b, then a / c > b / c. And, if a < b, then a / c < b / c. –If you divide each side of an inequality by a negative number, the inequality symbol changes direction. –For any real numbers a and b and any negative real number c, if a > b, then a / c b / c. –This property also holds for inequalities ≤ and ≥.

21. Example 7 Solve each inequality. a.12k ≥ 60 b.-8q < 136 12 k ≥ 5 {k|k ≥ 5} -8 q > -17 {q|q > -17}

22. Check Your Progress Choose the best answer for the following. A.Solve 15p < 60. A.{p|p < 4} B.{p|p < 45} C.{p|p < 75} D.{p|p > 4} 15 15p < 60

23. Check Your Progress Choose the best answer for the following. B.Solve -4z > 64. A.{z|z < 16} B.{z|z < -16} C.{z|z > -16} D.{z|z > 16} -4 -4z > 64