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Solving Multistep Inequalities

Solving Multistep Inequalities. 10-4. Pre-Algebra. 7. 3. 11. 8. 16. 16. x = – . Warm Up. Solve. 1. 6 x + 36 = 2 x 2. 4 x – 13 = 15 + 5 x 3. 5( x – 3) = 2 x + 3 4. + x = . x = –9. x = –28. x = 6.

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Solving Multistep Inequalities

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  1. Solving Multistep Inequalities 10-4 Pre-Algebra

  2. 7 3 11 8 16 16 x = – Warm Up Solve. 1.6x + 36 = 2x 2. 4x – 13 = 15 + 5x 3. 5(x – 3) = 2x + 3 4. + x = x = –9 x = –28 x = 6

  3. Learn to solve two-step inequalities and graph the solutions of an inequality on a number line.

  4. Solving a multistep inequality uses the same inverse operations as solving a multistep equation. Multiplying or dividing the inequality by a negative number reverses the inequality symbol.

  5. 1 2 3 4 5 6 7 12 4x > 4 4 Example Solve and graph. A. 4x + 1 > 13 4x + 1 > 13 – 1– 1Subtract 1 from both sides. 4x > 12 Divide both sides by 4. x > 3

  6. < -7 -6 -5 -4 -3 -2 -1 3x – 15 3 3 Example B. –7 < 3x + 8 –7 < 3x + 8 – 8– 8Subtract 8 from both sides. –15 < 3x Divide both sides by 3. –5 < x

  7. -6 -5 -4 -3 -2 -1 0 18 –9x –9 –9 Example C. -9x + 7  25 –9x + 7  25 – 7– 7Subtract 7 from both sides. –9x 18 Divide each side by –9; change  to . x–2

  8. 1 2 3 4 5 6 7 10 5x > 5 5 Try This Solve and graph. A. 5x + 2 > 12 5x + 2 > 12 – 2– 2Subtract 2 from both sides. 5x > 10 Divide both sides by 5. x > 2

  9. < -7 -6 -5 -4 -3 -2 -1 2x – 14 2 2 Try This B. –5 < 2x + 9 –5 < 2x + 9 – 9– 9Subtract 9 from both sides. –14 < 2x Divide both sides by 2. –7 < x

  10. -6 -5 -4 -3 -2 -1 0 16 –4x –4 –4 Try This C. -4x + 2  18 –4x + 2  18 – 2– 2Subtract 2 from both sides. –4x 16 Divide each side by –4; change  to . x–4

  11. -8 -7 -6 -5 -4 -3 -2 –36 6x < 6 6 Example Solve and graph. A. 10x + 21 – 4x < –15 10x + 21 – 4x < –15 6x + 21 < –15 Combine like terms. – 21– 21Subtract 21 from both sides. 6x < –36 Divide both sides by 6. x < –6

  12. B. +  +  20( + ) 20( ) 2x 2x 2x 2x 3 3 3 3 9 9 9 9 5 5 5 5 4 4 4 4 10 10 10 10 20( ) + 20( ) 20( ) Example Multiply by LCD, 20. 8x + 15 18 – 15– 15Subtract 15 from both sides. 8x 3

  13. Divide both sides by 8. x  8x 3 3 8 8 8 0 1 3 8 Example Continued 8x 3

  14. -1 0 1 2 3 4 5 3x 9 > 3 3 Example C. 8x + 8 > 11x – 1 8x + 8 > 11x– 1 – 8x– 8xSubtract 8x from both sides. 8 > 3x– 1 +1 +1 Add 1 to each side. 9 > 3x Divide both sides by 3. 3 > x

  15. -8 -7 -6 -5 -4 -3 -2 –40 10x < 10 10 Try This Solve and graph. A. 15x + 30 – 5x < –10 15x + 30 – 5x < –10 10x + 30 < –10 Combine like terms. – 30– 30Subtract 30 from both sides. 10x < –40 Divide both sides by 10. x < –4

  16. B. +  +  20( + ) 20( ) 3x 3x 3x 3x 1 1 1 1 5 5 5 5 5 5 5 5 4 4 4 4 10 10 10 10 20( ) + 20( ) 20 ( ) Try This Multiply by LCD, 20. 12x + 5 10 – 5– 5Subtract 5 from both sides. 12x 5

  17. x  Divide both sides by 12. 12x 5 5 12 12 12 0 5 12 Try This Continued 12x 5

  18. -1 0 1 2 3 4 5 4x 4 > 4 4 Try This C. 4x + 3 > 8x – 1 4x + 3 > 8x– 1 – 4x– 4xSubtract 4x from both sides. 3 > 4x– 1 +1 +1 Add 1 to each side. 4 > 4x Divide both sides by 4. 1 > x

  19. Example: Business Application A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit? Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost. R > C

  20. Example Continued The revenue from selling x bumper stickers at $1.25 each is 1.25x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or 55 + 0.15x. Substitute the expressions for R and C. Let x represent the number of bumper stickers sold. Fixed cost is $55. Unit cost is 15 cents. 1.25x > 55 + 0.15x

  21. 55 > 1.10x 1.10 1.10 Example Continued 1.25x > 55 + 0.15x Subtract 0.15x from both sides. – 0.15x– 0.15x 1.10x > 55 Divide both sides by 1.10. x > 50 The Spanish club must sell more than 50 bumper stickers to make a profit.

  22. Try This A school’s Spanish club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit? Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost. R > C

  23. Try This Continued The revenue from selling x bumper stickers at $2.50 each is 2.5x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or 45 + 0.25x. Substitute the expressions for R and C. Let x represent the number of bumper stickers sold. Fixed cost is $45. Unit cost is 25 cents. 2.5x > 45 + 0.25x

  24. 45 > 2.25x 2.25 2.25 Try This Continued 2.5x > 45 + 0.25x Subtract 0.25x from both sides. – 0.25x– 0.25x 2.25x > 45 Divide both sides by 2.25. x > 20 The Spanish club must sell more than 20 bumper stickers to make a profit.

  25. 2 3 3 8 1 4 1 2 3 4 5 6 7 -10 -9 -8 -7 -6 -5 -4 0 -18 -17 -16 -15 -14 -13 -12 1 2 w 3 8 Lesson Quiz: Part 1 Solve and graph. 1. 4x – 6 > 10 2. 7x + 9 < 3x – 15 3.w – 3w < 32 4.w +  x > 4 x < –6 w > –16

  26. Lesson Quiz: Part 2 5. Antonio has budgeted an average of $45 a month for entertainment. For the first five months of the year he has spent $48, $39, $60, $48, and $33. How much 1 can Antonio spend in the sixth month without exceeding his average budget? no more than $42

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