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Solve inequalities that contain variable terms on both sides.

Objective. Solve inequalities that contain variable terms on both sides. Example 1A: Solving Inequalities with Variables on Both Sides. Solve the inequality and graph the solutions. y ≤ 4 y + 18. Example 1B: Solving Inequalities with Variables on Both Sides.

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Solve inequalities that contain variable terms on both sides.

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

  2. Example 1A: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. y ≤ 4y + 18

  3. Example 1B: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. 4m – 3 < 2m + 6

  4. Check It Out! Example 1a Solve the inequality and graph the solutions. 4x ≥ 7x + 6

  5. Check It Out! Example 1b Solve the inequality and graph the solutions. 5t + 1 < –2t – 6

  6. 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 $36 per window, and the price includes power-washing the siding. 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.

  7. Home Cleaning Company siding charge Power Clean cost per window is less than # of windows. # of windows times $12 per window plus times Example 2 Continued 312 + 12 • w < 36 • w

  8. Example 3A: Simplify Each Side Before Solving Solve the inequality and graph the solutions. 2(k – 3) > 6 + 3k – 3

  9. Example 3B: Simplify Each Side Before Solving Solve the inequality and graph the solution. 0.9y ≥ 0.4y – 0.5

  10. Check It Out! Example 3a Solve the inequality and graph the solutions. 5(2 – r) ≥ 3(r – 2)

  11. Check It Out! Example 3b Solve the inequality and graph the solutions. 0.5x – 0.3 + 1.9x < 0.3x + 6

  12. There are special cases of inequalities called identities and contradictions.

  13. –2x –2x Example 4A: Identities and Contradictions Solve the inequality. 2x – 7 ≤ 5 + 2x 2x – 7 ≤ 5 + 2x Subtract 2x from both sides.  –7 ≤ 5 True statement. The inequality 2x − 7 ≤ 5 + 2x is an identity. All values of x make the inequality true. Therefore, all real numbers are solutions.

  14. –6y –6y Example 4B: Identities and Contradictions Solve the inequality. 2(3y – 2) – 4 ≥ 3(2y + 7) Distribute 2 on the left side and 3 on the right side. 2(3y – 2) – 4 ≥ 3(2y + 7) 2(3y) – 2(2) – 4 ≥ 3(2y) + 3(7) 6y – 4 – 4 ≥ 6y + 21 6y – 8 ≥ 6y + 21 Subtract 6y from both sides.  –8 ≥ 21 False statement. No values of y make the inequality true. There are no solutions.

  15. Check It Out! Example 4a Solve the inequality. 4(y – 1) ≥ 4y + 2

  16. Check It Out! Example 4b Solve the inequality. x – 2 < x + 1

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

  18. Lesson Quiz: Part II Solve each inequality. 4. 2y – 2 ≥ 2(y + 7) 5. 2(–6r – 5) < –3(4r + 2)

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