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**Solving Multistep Inequalities**

10-4 Pre-Algebra

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**Warm Up Solve. 1. 6x + 36 = 2x 2. 4x – 13 = 15 + 5x**

7 8 3 16 11 16 x = –

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

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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.

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**Example Solve and graph. A. 4x + 1 > 13 4x + 1 > 13**

– 1 – 1 Subtract 1 from both sides. 4x > 12 4x 4 > 12 Divide both sides by 4. x > 3

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**Example B. –7 < 3x + 8 – 8 – 8 Subtract 8 from both sides.**

– 15 3 < 3x Divide both sides by 3. –5 < x

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Example C. -9x + 7 25 –9x + 7 25 – 7 – 7 Subtract 7 from both sides. –9x 18 –9x –9 18 Divide each side by –9; change to . x –2

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**Try This Solve and graph. A. 5x + 2 > 12 5x + 2 > 12**

– 2 – 2 Subtract 2 from both sides. 5x > 10 5x 5 > 10 Divide both sides by 5. x > 2

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**Try This B. –5 < 2x + 9 – 9 – 9 Subtract 9 from both sides.**

– 14 2 < 2x Divide both sides by 2. –7 < x

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Try This C. -4x + 2 18 –4x + 2 18 – 2 – 2 Subtract 2 from both sides. –4x 16 –4x –4 16 Divide each side by –4; change to . x –4

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**Example Solve and graph. A. 10x + 21 – 4x < –15**

6x < – Combine like terms. – – 21 Subtract 21 from both sides. 6x < –36 6x 6 < –36 Divide both sides by 6. x < –6

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**Example B. + 2x 5 3 4 9 10 + 2x 5 3 4 9 10 20( + ) 20( ) 2x 5 3**

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

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**Example Continued 8x 3 8x 8 3 Divide both sides by 8. x 3 8 3 8**

3 8

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**Example C. 8x + 8 > 11x – 1 8x + 8 > 11x – 1**

– 8x – 8x Subtract 8x from both sides. 8 > 3x – 1 Add 1 to each side. 9 > 3x 9 3 > 3x Divide both sides by 3. 3 > x

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**Try This Solve and graph. A. 15x + 30 – 5x < –10**

10x < – Combine like terms. – – 30 Subtract 30 from both sides. 10x < –40 10x 10 < –40 Divide both sides by 10. x < –4

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**Try This B. + 3x 5 1 4 10 + 3x 5 1 4 10 20( + ) 20( ) 3x 5 1 4**

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

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**Try This Continued 12x 5 12x 12 5 Divide both sides by 12. x 5**

5 12

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**Try This C. 4x + 3 > 8x – 1 4x + 3 > 8x – 1**

– 4x – 4x Subtract 4x from both sides. 3 > 4x – 1 Add 1 to each side. 4 > 4x 4 > 4x Divide both sides by 4. 1 > x

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**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

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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 x. Substitute the expressions for R and C. 1.25x > x Let x represent the number of bumper stickers sold. Fixed cost is $55. Unit cost is 15 cents.

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**Example Continued 1.25x > 55 + 0.15x**

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

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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

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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 x. Substitute the expressions for R and C. 2.5x > x Let x represent the number of bumper stickers sold. Fixed cost is $45. Unit cost is 25 cents.

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**Try This Continued 2.5x > 45 + 0.25x**

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

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**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 2 3 1 4 1 2 w 3 8 3 8

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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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