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Warm Up: Solving Equations and Inequalities in California Standards Lesson

This warm-up presentation prepares students for solving multistep problems involving linear equations and inequalities in one variable, according to California Standards. The presentation covers solving equations and graphing inequalities, with an emphasis on multiplying or dividing by positive and negative numbers.

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Warm Up: Solving Equations and Inequalities in California Standards Lesson

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Solve each equation. 1. –5a = 30 2. –10 –6 3. 4. Graph each inequality. 5. x ≥ –10 6.x < –3

  3. California Standards Preparation for 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

  4. Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number. The rules on the next slide show the properties of inequality for multiplying or dividing by a positive number. The rules for multiplying or dividing by a negative number appear later in this lesson.

  5. –8 –2 –10 –6 –4 0 2 4 6 8 10 Additional Example 1A: Multiplying or Dividing by a Positive Number Solve the inequality and graph the solutions. 7x > –42 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. 1x > –6 x > –6 The solution set is {x: x > –6}.

  6. 3(2.4) ≤ 3 0 2 4 6 8 10 14 20 12 18 16 Additional Example 1B: Multiplying or Dividing by a Positive Number Solve the inequality and graph the solutions. Since m is divided by 3, multiply both sides by 3 to undo the division. The solution set is {m:m ≥ 7.2}. 7.2 ≤ m (or m ≥ 7.2)

  7. Since r is multiplied by , multiply both sides by the reciprocal of . 0 2 4 6 8 10 14 20 12 18 16 Additional Example 1C: Multiplying or Dividing by a Positive Number Solve the inequality and graph the solutions. r < 16 The solution set is {r:r < 16}.

  8. 14 0 2 4 6 8 10 20 12 18 16 Check It Out! Example 1a Solve the inequality and graph the solutions. Check your answer. 4k > 24 Since k is multiplied by 4, divide both sides by 4. k > 6 The solution set is {k:k > 6}.

  9. Check the endpoint, 6. Check a number greater than 6. 4(k) = 24 4(k) > 24 4(6) 24 4(7) > 24  24 24 28 > 24  Check It Out! Example 1a Continued Solve the inequality and graph the solutions. Check your answer. 4k > 24 Check

  10. –10 ≥ q or q ≤ –10 –15 –10 –5 0 5 15 Check It Out! Example 1b Solve the inequality and graph the solutions. Check your answer. –50 ≥ 5q Since q is multiplied by 5, divide both sides by 5. The solution set is {q:q ≤ –10}.

  11. Check a number less than or equal to –10. Check the endpoint, –10. –50 = 5(q) –50 ≥ 5(q) –50 5(–10) –50 ≥ 5(–11)  –50 –50  –50 ≥ –55 Check It Out! Example 1b Continued Solve the inequality and graph the solutions. Check your answer. –50 ≥ 5q Check

  12. Since g is multiplied by , multiply both sides by the reciprocal of . 20 25 30 35 15 40 Check It Out! Example 1c Solve the inequality and graph the solutions. Check your answer. g > 36 The solution set is {g:g > 36}. 36

  13. Check a number greater than 36. > 27 27   > 30 27 Check It Out! Example 1c Continued Solve the inequality and graph the solutions. Check your answer. Check Check the endpoint, 36.

  14. –b –a 0 a b a < b b > –a Multiply both sides by –1. Multiply both sides by –1. –a –b –b a You can tell from the number line that –a > –b. You can tell from the number line that –b < a. What happens when you multiply or divide both sides of an inequality by a negative number? Look at the number line below. Notice that when you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol.

  15. Caution! Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

  16. –7 –14 –12 –8 –2 –10 –6 –4 0 2 4 6 Additional Example 2A: Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions. –12x > 84 Since x is multiplied by –12, divide both sides by –12. Change > to <. x < –7

  17. Since x is divided by –3, multiply both sides by –3. Change to . 10 14 16 18 20 22 24 26 28 30 12 Additional Example 2B: Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions. 24  x (or x  24)

  18. –8 –2 –10 –6 –4 0 2 4 6 8 10 Check It Out! Example 2a Solve the inequality and graph the solutions. Check your answer. 10 ≥ –x Multiply both sides by –1 to make x positive. Change  to . –1(10) ≤ –1(–x) –10 ≤ x

  19. Check Check a number greater than 10. Check the endpoint, –10. 10 = –x 10 ≥ –x 10 –(–10) 10 ≥ –(11)  10 10  10 ≥ –11 Check It Out! Example 2a Continued Solve the inequality and graph the solutions. Check your answer. 10 ≥ –x

  20. –17 –4 –12 –8 0 4 8 12 16 –16 –20 20 Check It Out! Example 2b Solve the inequality and graph the solutions. Check your answer. 4.25 > –0.25h Since h is multiplied by –0.25, divide both sides by –0.25. Change > to <. –17 < h

  21. Check a number greater than –17. Check 4.25 > –0.25(h) Check the endpoint, –17. > 4.25 –0.25(–16) 4.25 = –0.25(h)  4.25 4 > 4.25 –0.25(–17)  4.25 4.25 Check It Out! Example 2b Continued Solve the inequality and graph the solutions. Check your answer. 4.25 > –0.25h

  22. number of tubes is at most times $20.00. $4.30 4.30 20.00 ≤ p • Additional Example 3:Application Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy? Let p represent the number of tubes of paint that Jill can buy.

  23. Additional Example 3 Continued Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy? 4.30p ≤ 20.00 Since p is multiplied by 4.30, divide both sides by 4.30. The symbol does not change. p ≤ 4.65… Since Jill can buy only whole numbers of tubes, she can buy 0, 1, 2, 3, or 4 tubes of paint.

  24. number of servings is at most times 128 oz 10 oz 128 10 ≤ g • Check It Out! Example 3 A pitcher holds 128 ounces of juice. What are the possible numbers of 10-ounce servings that one pitcher can fill? Let g represent the number of servings of juice the pitcher can contain.

  25. Check It Out! Example 3 Continued A pitcher holds 128 ounces of juice. What are the possible numbers of 10-ounce servings that one pitcher can fill? 10g ≤ 128 Since g is multiplied by 10, divide both sides by 10. The symbol does not change. g ≤ 12.8 The pitcher can fill 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 servings.

  26. Lesson Quiz Solve each inequality and graph the solutions. 1. 8x < –24 x < –3 2. –5x ≥30 x ≤ –6 4. 3. x > 20 x ≥ 6 5. A soccer coach plans to order more shirts for her team. Each shirt costs $9.85. She has $77 left in her uniform budget. What are the possible numbers of shirts she can buy? 0, 1, 2, 3, 4, 5, 6, or 7 shirts

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