1. 3-5 Solving Inequalities with Variables on Both Sides Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2. 3-5 Solving Inequalities with Variables on Both Sides Warm Up Solve each equation. 1. 2x = 7x + 15 2. 5. Solve and graph 5(2 – b) > 5 2. 3. 2(3z + 1) = – 2(z + 3) 4. 3(p – 1) = 3p + 2 x = –3 b < –3 –5 –3–2–1 –4 0 –6 3y – 21 = 4 – 2yy = 5 z = –1 no solution

3. 3-5 Solving Inequalities with Variables on Both Sides MA.912.A.3.5 Symbolically represent and solve multi-step and real-world applications that involve linear…inequalities. Also MA.912.A.3.4, MA.912.A.10.3. Sunshine State Standards

4. 3-5 Solving Inequalities with Variables on Both Sides Solve inequalities that contain variable terms on both sides. Objective

5. 3-5 Solving Inequalities with Variables on Both Sides Some inequalities have variable terms on both sides of the inequality symbol. You can solve these inequalities like you solved equations with variables on both sides. Use the properties of inequality to “ collect ” all the variable terms on one side and all the constant terms on the other side.

6. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 1A: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. y ≤ 4y + 18 –y 0 ≤ 3y + 18 –18 – 18 –18 ≤ 3y –6 ≤ y (or y  –6) To collect the variable terms on one side, subtract y from both sides. Since 18 is added to 3y, subtract 18 from both sides to undo the addition. Since y is multiplied by 3, divide both sides by 3 to undo the multiplication. –10 –8 –6–4 –2 0246810

7. 3-5 Solving Inequalities with Variables on Both Sides 4m – 3 < 2m + 6 To collect the variable terms on one side, subtract 2m from both sides. –2m – 2m 2m – 3 < + 6 Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction + 3 2m < 9 Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 4 5 6 Additional Example 1B: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions.

8. 3-5 Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. Check It Out! Example 1a 4x ≥ 7x + 6 –7x –3x ≥ 6 x ≤ –2 To collect the variable terms on one side, subtract 7x from both sides. Since x is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤. –10 –8 –6–4 –2 0246810

9. 3-5 Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. Check It Out! Example 1b To collect the variable terms on one side, subtract 3 from both sides. Subtract one-fourth t from both sides. –3–3 –3–3

10. 3-5 Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. Check It Out! Example 1b Continued Divide both sides by ten-fourths. –5 –4 –3–2 –1 01234 5

11. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 2: Business Application The Home Cleaning Company charges $312 to power-wash the siding of a house plus $12 for each window. Power Clean charges $156, to power-washing the siding plus $24 per window. How many windows must a house have to make the total cost from The Home Cleaning Company less expensive than Power Clean? Let w be the number of windows.

12. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 2 Continued 312 + 12 w < 156 + 24 w −156 – 12w −156 –12w 156 < 12w 13 < w The Home Cleaning Company is less expensive for houses with more than 13 windows. To collect like terms, subtract 12w and 156 from both sides. Since w is multiplied by 12, divide both sides by 12 to undo the multiplication. Home Cleaning Company siding charge plus $12 per window # of windows is less than Power Clean siding charge # of windows. times $24 per window plus

13. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 2 A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. For how many flyers is the cost at A-Plus Advertising less than the cost of Print and More? Let f represent the number of flyers printed. 24 + 0.10 f < 0.25 f plus $0.10 per flyer is less than # of flyers. A-Plus Advertising fee of $24 Print and More’s cost per flyer # of flyers times

14. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 2 Continued 24 + 0.10f < 0.25f –0.10f 24 < 0.15f 160 < f To collect the variable terms, subtract 0.10f from both sides. Since f is multiplied by 0.15, divide both sides by 0.15 to undo the multiplication. More than 160 flyers must be delivered to make A-Plus Advertising the lower cost company.

15. 3-5 Solving Inequalities with Variables on Both Sides You may need to simplify one or both sides of an inequality before solving it. Look for like terms to combine and places to use the Distributive Property.

16. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 3A: Simplify Each Side Before Solving Solve the inequality and graph the solutions. 2(k – 3) > 6 + 3k – 3 2(k – 3) > 3 + 3k Distribute 2 on the left side of the inequality. 2k + 2(–3) > 3 + 3k 2k – 6 > 3 + 3k –2k – 2k –6 > 3 + k To collect the variable terms, subtract 2k from both sides. –3 –9 > k Since 3 is added to k, subtract 3 from both sides to undo the addition.

17. 3-5 Solving Inequalities with Variables on Both Sides –9 > k –12–9–6–303 Additional Example 3A Continued Solve the inequality and graph the solutions.

18. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 3B: Simplify Each Side Before Solving Solve the inequality and graph the solution. 3.2y − 2.3y ≥ 0.4y – 0.5 –0.4y 0.5y ≥ – 0.5 0.5 y ≥ –1 To collect the variable terms, subtract 0.4y from both sides. Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication. –5 –4 –3–2 –1 01234 5 3.2y - 2.3y ≥ 0.4 y - 0.5 0.9y ≥ 0.4y – 0.5 Combine y terms.

19. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 3a Solve the inequality and graph the solutions. 5(2 – r) ≥ 3(r – 2) 5(2) – 5(r) ≥ 3(r) + 3(–2) 10 – 5r ≥ 3r – 6 +6 16 − 5r ≥ 3r + 5r +5r 16 ≥ 8r Distribute 5 on the left side of the inequality and distribute 3 on the right side of the inequality. Since 6 is subtracted from 3r, add 6 to both sides to undo the subtraction. Since 5r is subtracted from 16 add 5r to both sides to undo the subtraction.

20. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 3a Continued –6 –202 –4 4 16 ≥ 8r Since r is multiplied by 8, divide both sides by 8 to undo the multiplication. 2 ≥ r Solve the inequality and graph the solutions.

21. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 3b Solve the inequality and graph the solutions. 0.5x – 0.3 + 1.9x < 0.3x + 6 2.4x – 0.3 < 0.3x + 6 + 0.3 2.4x < 0.3x + 6.3 –0.3x 2.1x < 6.3 Since 0.3 is subtracted from 2.4x, add 0.3 to both sides. Since 0.3x is added to 6.3, subtract 0.3x from both sides. x < 3 Since x is multiplied by 2.1, divide both sides by 2.1. Simplify. 2.4x – 0.3 < 0.3x + 6

22. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 3b Continued x < 3 –5 –4 –3–2 –1 01234 5 Solve the inequality and graph the solutions.

23. 3-5 Solving Inequalities with Variables on Both Sides Some inequalities are true no matter what value is substituted for the variable. For these inequalities, all real numbers are solutions. Some inequalities are false no matter what value is substituted for the variable. These inequalities have no solutions. If both sides of an inequality are fully simplified and the same variable term appears on both sides, then the inequality has all real numbers as solutions or it has no solutions. Look at the other terms in the inequality to decide which is the case.

24. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 4A: All Real Numbers as Solutions or No Solutions Solve the inequality. 2x – 7 ≤ 5 + 2x The same variable term (x) appears on both sides. Look at the other terms. For any number x, subtracting 7 will always result in a lesser number than adding 5. All values of x make the inequality true. All real numbers are solutions.

25. 3-5 Solving Inequalities with Variables on Both Sides Additional Example 4B: All Real Numbers as Solutions or No Solutions Solve the inequality. The same variable term (y) appears on both sides. Look at the other terms. 2 (3y – 2) – 4 ≥ 3(2y + 7) 2(3y−2) − 4 ≥ 3(2y +7) 6y − 4 − 4 ≥ 6y + 21 Distribute 2 on the left side and 3 on the right side of the inequality. Add -4’s on the left side. 6y − 8 ≥ 6y + 21

26. 3-5 Solving Inequalities with Variables on Both Sides For any number y, subtracting 8 will always result in a lesser number than adding 21. No values of y make the inequality true. There are no solutions. Additional Example 4B Continued 6y − 8 ≥ 6y + 21 Solve the inequality. 2 (3y – 2) – 4 ≥ 3(2y + 7)

27. 3-5 Solving Inequalities with Variables on Both Sides 4(y – 1) ≥ 4y + 2 4(y) + 4(–1) ≥ 4y + 2 4y – 4 ≥ 4y + 2 Distribute 4 on the left side. Check It Out! Example 4a Solve the inequality. –4y –4 ≥ 2 Subtract 4y from both sides. False statement.  No values of y make the inequality true. There are no solutions.

28. 3-5 Solving Inequalities with Variables on Both Sides Check It Out! Example 4b Solve the inequality. x – 2 < x + 1 The same variable term (x) appears on both sides. Look at the other terms. For any number x, subtracting 2 will always result in a lesser number than adding 1. All values of x make the inequality true. All real numbers are solutions. x – 2 < x + 1

29. 3-5 Solving Inequalities with Variables on Both Sides Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

30. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. t < 5t + 24t > –6 2. 5x – 9 ≤ 4.1x – 81x ≤ –80 b < 133. 4b + 4(1 – b) > b – 9

31. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz: Part II 4. Rick bought a photo printer and supplies for $186.90, which will allow him to print photos for $0.29 each. A photo store charges $0.55 to print each photo. How many photos must Rick print before his total cost is less than getting prints made at the photo store? Rick must print more than 718 photos.

32. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz: Part III Solve each inequality. 5. 2y – 2 ≥ 2(y + 7) no solutions 6. 2(–6r – 3) < –3(4r + 2) all real numbers

33. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz for Student Response Systems 1. Identify the solution for the inequality. A. D. a > – 9 B. a < 5 C. a > 5 a < – 9 a < 6a + 45

34. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz for Student Response Systems 2. Identify the solution for the inequality. A. D. t ≤ – 6 B. t ≥ – 60 C. t ≤ – 60 t ≥ – 6 6t + 4 ≤ 5.4t − 32

35. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz for Student Response Systems 3. Identify the solution for the inequality. A. D. y < 10 B. y < 15 C. y < – 15 y < – 10 7y + 7(3 − y) > 2y − 9

36. 3-5 Solving Inequalities with Variables on Both Sides 4. John bought a computer scanner and supplies for $215.70, which will allow him to scan images for $0.34 each. A computer center charges $0.59 to scan each image. How many images must John scan before his total cost is less than getting scanned images at the computer center? Lesson Quiz for Student Response Systems A. John must scan more than 863 images. D. B. C. John must scan more than 788 images. John must scan more than 862 images. John must scan more than 768 images.

37. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz for Student Response Systems 5. Solve the inequality. 3y − 4 ≥ 3(y + 4) A. all real numbers B. no solutions

38. 3-5 Solving Inequalities with Variables on Both Sides Lesson Quiz for Student Response Systems 6. Solve the inequality. – 2(8x − 2) ≤ 4( – 4x + 1) A. all real numbers B. no solutions