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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up Compare. Use < or >. 1. 5 7 2. –3 –4 3. 2.5 –2.7 4. –8 –7 Solve. 5. 4 + y = 16 6. m – 7 = 14 7. – 3 = 8 + w 8. 7 = t + 10. >. <. >. <. 12. 21. –11. –3. California

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Compare. Use < or >. 1. 5 72. –3 –4 3. 2.5 –2.7 4. –8 –7 Solve. 5. 4 + y = 16 6. m – 7 = 14 7. –3 = 8 + w 8. 7 = t + 10 > < > < 12 21 –11 –3

  3. California Standards AF4.0 Students solve simple linear equations and inequalities over the rational numbers.

  4. When you ADD OR SUBTRACT the same number on both sides of an inequality, the resulting statement will still be true. –2 < 5 +7 +7 5 < 12 You can find solution sets of inequalities the same way you find solutions of equations, by ISOLATING THE VARIABLE

  5. –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Solve and graph the inequality. x + 3 > –5 x + 3 > –5 Since 3 is added x, subtract 3 from both sides. –3 –3 x > –8

  6. Check According to the graph –4 should be a solution and –9 should not be a solution. x + 3 > –5 x + 3 > –5 Substitute –4 for x. Substitute –9 for x. ? ? –4 + 3 > –5 –9 + 3 > –5 ? ? –1 > –5 –6 > –5  So –4 is a solution. So –9 is not a solution.

  7. –1 0 1 2 3 4 5 6 7 8 9 10 11 12 Solve and graph the inequality. m –4 ≥ –2 m – 4 ≥ –2 Since 4 is subtracted from m, add 4 to both sides. + 4 + 4 m ≥ 2

  8. –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Solve and graph the inequality. r +3 ≤ –3 r +3≤ –3 Since 3 is added to r, subtract 3 from both sides. – 3 –3 r ≤ –6

  9. –7 -6 –5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Solve and graph the inequality. 34 14 5 > n + 1 34 14 5 > n + 1 Since 1¼ is added to n, subtract 1¼ from both sides. 14 14 – 1 – 1 12 4 >n

  10. –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Solve and graph the inequality. x + 4 > –2 x + 4 > –2 Since 4 is added x, subtract 4 from both sides. –4 –4 x > –6

  11. Check According to the graph 2 should be a solution and –8 should not be a solution. x + 4 > –2 x + 4 > –2 Substitute 2 for x. Substitute –8 for x. ? ? 2 + 4 > –2 –8 + 4 > –2 ? ? 6 > –2 –4 > –2  So 2 is a solution. So –8 is not a solution.

  12. –1 0 1 2 3 4 5 6 7 8 9 10 11 12 Solve and graph the inequality. w –8 ≥ –3 w –8 ≥ –3 Since 8 is subtracted from w, add 8 to both sides. + 8 + 8 w ≥ 5

  13. –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Solve and graph the inequality. c +6 ≤ –1 c +6≤ –1 Since 6 is added to c, subtract 6 from both sides. – 6 – 6 c ≤ –7

  14. –7 -6 –5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Solve and graph the inequality. 23 13 3 > n + 1 23 13 3 > n + 1 13 Since 1 is added to n, subtract 1 from both sides. 13 13 – 1 – 1 13 13 2 >n

  15. Additional Example 2: Sports Application While training for a race, Ann’s goal is to run at least 3.5 miles each day. She has already run 1.8 miles today. Write and solve an inequality to find out how many more miles she must run today. Let m = the number of additional miles. 1.8 miles plus additional miles is at least3.5 miles. 1.8+m≥ 3.5 1.8 + m ≥ 3.5 Since 1.8 is added to m, subtract 1.8 from both sides. –1.8 –1.8 m ≥ 1.7 Ann should run at least 1.7 more miles.

  16. ? ? 1.8 + m ≥ 3.5 1.8 + 2 ≥ 3.5 3.8 ≥ 3.5 1.8 + m ≥ 3.5 1.8 + 1 ≥ 3.5 2.8 ≥ 3.5 ? ? Additional Example 2 Continued Check 2 is greater than 1.7. Substitute 2 for m. 1 is less than 1.7. Substitute 1 for m. x

  17. Check It Out! Example 2 Tim’s company produces recycled paper. They produce 60.5 lb of paper each day. They have already produced at least 20.2 lb today. Write and solve an inequality to find out how many more pounds Tim’s company must produce. Let p = the number of additional pounds of paper. 20.2 lbs plus additional pounds is at least 60.5 lb. 20.2+p≥ 60.5 20.2 + p ≥ 60.5 Since 20.2 is added to p, subtract 20.2 from both sides. –20.2 –20.2 p ≥ 40.3 Tim’s company should produce at least 40.3 lb more of paper.

  18. ? ? 20.2 + p ≥ 60.5 20.2 + 41 ≥ 60.5 61.2 ≥ 60.5 20.2 + p ≥ 60.5 20.2 + 40 ≥ 60.5 60.2 ≥ 60.5 ? ? Check It Out! Example 2 Continued Check 41 is greater than 40.3. Substitute 41 for p. 40 is less than 40.3. Substitute 40 for p. x

  19. g < 4 -1 0 1 2 3 4 5 s ≥ –1 • -4 -3 -2 -1 0 1 2 -4 -3 -2 -1 0 1 2 x ≥ 0 • 45 y < 1/5 2/5 3/5 4/5 1 1 1/5 Lesson Quiz: Part I Solve and graph each inequality. 1.g – 7 < –3 2. 5 + s ≥ 4 3. –5.1 ≤ x – 5.1 15 4. 3 + y > 4

  20. Lesson Quiz: Part II 5. Tasha is folding letters for a fundraiser. She knows there are at least 300 letters, and she has already folded 125 of them. Write and solve an inequality to show how many more letters she must fold. 125 + x ≥ 300; x ≥ 175

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