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Inequalities with Variable Terms Presentation

Learn to solve multi-step inequalities with variable terms on both sides using real-world examples. Practice equations and inequalities.

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Inequalities with Variable Terms Presentation

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  1. Warm Up Lesson Presentation Lesson Quiz

  2. –4 –3 –2 –1 –5 –6 0 Warm Up Solve each equation. 1. 2x = 7x + 15 2. x = –3 3y – 21 = 4 – 2y y = 5 z = –1 3. 2(3z + 1) = –2(z + 3) 4. 3(p –1) = 3p + 2 no solution b < –3 5. Solve and graph 5(2 –b) > 52.

  3. Sunshine State Standards MA.912.A.3.5 Symbolically represent and solve multi-step and real-world applications that involve linear…inequalities. AlsoMA.912.A.3.4, MA.912.A.10.3.

  4. Objective Solve inequalities that contain variable terms on both sides.

  5. 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. y ≤ 4y + 18 –y –y 0 ≤ 3y + 18 –18 – 18 –18 ≤ 3y –8 –10 –6 –4 0 2 4 6 8 10 –2 Additional Example 1A: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. y ≤ 4y + 18 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. –6 ≤ y (or y –6)

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

  8. 4x ≥ 7x + 6 –7x –7x x ≤ –2 –8 –10 –6 –4 0 2 4 6 8 10 –2 Check It Out! Example 1a Solve the inequality and graph the solutions. 4x ≥ 7x + 6 To collect the variable terms on one side, subtract 7x from both sides. –3x ≥ 6 Since x is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.

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

  10. –4 –1 5 –3 –2 0 1 2 3 4 –5 Check It Out! Example 1b Continued Solve the inequality and graph the solutions. Divide both sides by ten-fourths.

  11. 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. 312 + 12w < 156 + 24w −156 – 12w −156 –12w Additional Example 2 Continued Home Cleaning Company siding charge Power Clean siding charge is less than $24 per window # of windows. # of windows times $12 per window plus plus times 312 + 12 • w < 156 + 24 • w To collect like terms, subtract 12w and 156 from both sides. 156 < 12w Since w is multiplied by 12, divide both sides by 12 to undo the multiplication. 13 < w The Home Cleaning Company is less expensive for houses with more than 13 windows.

  13. $0.10 per flyer Print and More’s cost per flyer A-Plus Advertising fee of $24 # of flyers. # of flyers is less than times times plus 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

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

  15. 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. –2k –2k –3 –3 Additional Example 3A: Simplify Each Side Before Solving Solve the inequality and graph the solutions. 2(k – 3) > 6 + 3k – 3 Distribute 2 on the left side of the inequality. 2(k – 3) > 3 + 3k 2k+ 2(–3)> 3 + 3k 2k –6 > 3 + 3k To collect the variable terms, subtract 2k from both sides. –6 > 3 + k Since 3 is added to k, subtract 3 from both sides to undo the addition. –9 > k

  17. –12 –9 –6 –3 0 3 Additional Example 3A Continued Solve the inequality and graph the solutions. –9 > k

  18. –0.4y –0.4y 0.5y ≥ – 0.5 0.5y ≥–0.5 0.5 0.5 –4 –1 5 –3 –2 0 1 2 3 4 –5 Additional Example 3B: Simplify Each Side Before Solving Solve the inequality and graph the solution. 3.2y - 2.3y ≥ 0.4 y - 0.5 3.2y − 2.3y ≥ 0.4y – 0.5 Combine y terms. 0.9y ≥ 0.4y – 0.5 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. y ≥ –1

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

  20. –6 –4 –2 0 2 4 Check It Out! Example 3a Continued Solve the inequality and graph the solutions. 16 ≥ 8r Since r is multiplied by 8, divide both sides by 8 to undo the multiplication. 2 ≥ r

  21. + 0.3 + 0.3 –0.3x –0.3x 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 Simplify. Since 0.3 is subtracted from 2.4x, add 0.3 to both sides. 2.4x –0.3 < 0.3x + 6 2.4x < 0.3x + 6.3 Since 0.3x is added to 6.3, subtract 0.3x from both sides. 2.1x < 6.3 Since x is multiplied by 2.1, divide both sides by 2.1. x < 3

  22. –4 –1 5 –3 –2 0 1 2 3 4 –5 Check It Out! Example 3b Continued Solve the inequality and graph the solutions. x < 3

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

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

  26. Additional Example 4B Continued Solve the inequality. 2 (3y – 2) – 4 ≥ 3(2y + 7) 6y − 8 ≥ 6y+ 21 For any numbery, subtracting 8will always result in a lesser number than adding 21. No values of ymake the inequality true. There are no solutions.

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

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

  29. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  30. Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. t < 5t + 24 t > –6 2. 5x – 9 ≤ 4.1x –81 x ≤–80 3. 4b + 4(1 – b) > b – 9 b < 13

  31. 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. 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. Lesson Quiz for Student Response Systems 1. Identify the solution for the inequality. a < 6a + 45 a >–9 A. B. a <– 9 a >5 C. a <5 D.

  34. Lesson Quiz for Student Response Systems 2. Identify the solution for the inequality. 6t + 4 ≤ 5.4t − 32 A. t ≤–6 B. t ≥–6 t ≤–60 C. t ≥–60 D.

  35. Lesson Quiz for Student Response Systems 3. Identify the solution for the inequality. 7y + 7(3 − y) > 2y − 9 y <10 A. B. y <–10 y <–15 C. y <15 D.

  36. Lesson Quiz for Student Response Systems 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? A. John must scan more than 863 images. B. John must scan more than 862 images. John must scan more than 788 images. C. John must scan more than 768 images. D.

  37. Lesson Quiz for Student Response Systems 5. Solve the inequality. 3y − 4 ≥ 3(y + 4) A. all real numbers B. no solutions

  38. Lesson Quiz for Student Response Systems 6. Solve the inequality. –2(8x − 2) ≤ 4(–4x + 1) A. all real numbers B. no solutions

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