Page 510 #10-20 ANSWERS

Student Progress Chart Lesson Reflection

Today’s Learning Goal Assignment Learn to solve two-step inequalities and graph the solutions of an inequality on a number line.

Today’s Learning Goal Assignment Page 517 #14-26 Solve & Graph!

Solving Multistep Inequalities 10-4 Solving Multistep Inequalities Warm Up Problem of the Day Lesson Presentation Pre-Algebra

Solving Multistep Inequalities Pre-Algebra 10-4 Solving Multistep Inequalities 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 7 8 3 16 11 16 x = –

Find an integer x that makes the following two inequalities true: Problem of the Day Find an integer x that makes the following two inequalities true: 4 < x2 < 16 and x < 2.5 x = –3

Today’s Learning Goal Assignment 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.

Additional Example 1A: Solving Multistep Inequalities 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 1 2 3 4 5 6 7

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

Additional Example 1C: Solving Multistep Inequalities 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 -6 -5 -4 -3 -2 -1 0

– 2 – 2 Subtract 2 from both sides. Try This: Example 1A 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 1 2 3 4 5 6 7

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

– 2 – 2 Subtract 2 from both sides. Try This: Example 1C 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 -6 -5 -4 -3 -2 -1 0

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

Additional Example 2B: Solving Multistep Inequalities 5 3 4 9 10 +  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 – 15 Subtract 15 from both sides. 8x  3

Additional Example 2 Continued  8x 8 3 Divide both sides by 8. x  3 8 0 1 3 8

Additional Example 2C: Solving Multistep Inequalities C. 8x + 8 > 11x – 1 8x + 8 > 11x – 1 – 8x – 8x Subtract 8x from both sides. 8 > 3x – 1 +1 +1 Add 1 to each side. 9 > 3x 9 3 > 3x Divide both sides by 3. 3 > x -1 0 1 2 3 4 5

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

Try This: Example 2B B. +  3x 5 1 4 10 +  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 – 5 Subtract 5 from both sides. 12x  5

Try This: Example 2B Continued  12x 12 5 Divide both sides by 12. x  5 12 5 12

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

Additional Example 3: 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

Additional Example 3 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. 1.25x > 55 + 0.15x Let x represent the number of bumper stickers sold. Fixed cost is $55. Unit cost is 15 cents.

Additional Example 3 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: Example 3 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: Example 3 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. 2.5x > 45 + 0.25x Let x represent the number of bumper stickers sold. Fixed cost is $45. Unit cost is 25 cents.

Try This: Example 3 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 +  1 2 3 4 5 6 7 x > 4 -10 -9 -8 -7 -6 -5 -4 x < –6 -18 -17 -16 -15 -14 -13 -12 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