11-5 Solving Two-Step Inequalities Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Warm Up Solve. 1. 6x + 36 = 2x 2. 4x – 13 = x 3. 5(x – 3) = 2x x = x = –9 x = –28 x = 6 Course Solving Two-Step Inequalities x = –

Problem of the Day Find an integer x that makes the following two inequalities true: 4 < x 2 < 16 and x < 2.5 x = –3 Course Solving Two-Step Inequalities

Learn to solve two-step inequalities and graph the solutions of an inequality on a number line. Course Solving Two-Step Inequalities

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. Course Solving Two-Step Inequalities

Solve and graph. Additional Example 1A: Solving Two-Step Inequalities 4x + 1 > 13 – 1 – 1Subtract 1 from both sides. 4x > 12 4x4x 4 > 12 4 Divide both sides by 4. x > Course Solving Two-Step Inequalities

Course Solving Two-Step Inequalities If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. Remember!

Additional Example 1B: Solving Two-Step Inequalities –9x + 7  25 – 7 – 7Subtract 7 from both sides. –9x  18 –9x –9  18 –9 Divide each side by –9; change  to . x  –2x  – Course Solving Two-Step Inequalities Solve and graph.

Check It Out: Example 1A 5x + 2 > 12 – 2 – 2Subtract 2 from both sides. 5x > 10 5x5x 5 > 10 5 Divide both sides by 5. x > Course Solving Two-Step Inequalities

–4x + 2  18 – 2 – 2Subtract 2 from both sides. –4x  16 –4x –4  16 –4 Divide each side by –4; change  to . x  –4x  – Check It Out: Example 1B Course Solving Two-Step Inequalities

Additional Example 2: Solving Inequalities That Contain Fractions Multiply by LCD, 20. 8x + 15  18 – 15 – 15 Subtract 15 from both sides. 8x  3 Solve +  and graph the solution. 2x  2x2x ( + )  20 ( ) 2x2x ( ) + 20 ( )  20 ( ) 2x2x Course Solving Two-Step Inequalities Distributive Property.

Additional Example 2 Continued x  3 8  8x8x Divide both sides by 8. 8x  Course Solving Two-Step Inequalities

Check It Out: Example 2 Multiply by LCD, x + 5  10 – 5 – 5 Subtract 5 from both sides. 12x  5 Solve +  3x  3x3x ( + )  20 ( ) 3x3x ( ) + 20 ( )  20 ( ) 3x3x Course Solving Two-Step Inequalities Distributive Property.

Check It Out: Example 2 Continued x  5 12  12x 12 5 Divide both sides by x  Course Solving Two-Step Inequalities

Additional Example 3: School 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 Course Solving Two-Step Inequalities

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 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. Course Solving Two-Step Inequalities

Additional Example 3 Continued – 0.15x Subtract 0.15x from both sides. 1.10x > 55 x > 50 The Spanish club must sell more than 50 bumper stickers to make a profit. Divide both sides by x > x 1.10x > Course Solving Two-Step Inequalities

Check It Out: Example 3 R > C A school’s French 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 French club to make a profit, the revenue must be greater than the cost. Course Solving Two-Step Inequalities

Check It Out: 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 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. Course Solving Two-Step Inequalities

– 0.25x Subtract 0.25x from both sides. 2.25x > 45 x > 20 The French club must sell more than 20 bumper stickers to make a profit. Divide both sides by x > x 2.25x > Check It Out: Example 3 Continued Course Solving Two-Step Inequalities

Lesson Quiz: Part I Solve and graph. 1. 4x – 6 > x + 9 < 3x – w – 3w < w +  x < –6 x > 4 Insert Lesson Title Here w > – w w  Course Solving Two-Step Inequalities

Lesson Quiz: Part II 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 can Antonio spend in the sixth month without exceeding his average budget? no more than $42 Course Solving Two-Step Inequalities