 Solving Multistep Inequalities

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Solving Multistep Inequalities
10-4 Pre-Algebra

Warm Up Solve. 1. 6x + 36 = 2x 2. 4x – 13 = 15 + 5x
7 8 3 16 11 16 x = –

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

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.

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

Example B. –7 < 3x + 8 – 8 – 8 Subtract 8 from both sides.
– 15 3 < 3x Divide both sides by 3. –5 < x

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

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

Try This B. –5 < 2x + 9 – 9 – 9 Subtract 9 from both sides.
– 14 2 < 2x Divide both sides by 2. –7 < x

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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.

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.

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

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