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Solving Two-Step Inequalities and Graphing Solutions

In this lesson, we focus on solving two-step inequalities and graphing their solutions. Building on previous knowledge of two-step equations, students will apply the order of operations in reverse to isolate the variable and solve inequalities like 4x + 1 > 13. We will cover additional examples involving practical applications, such as calculating profits from selling bumper stickers for clubs. Emphasis will be placed on understanding when to reverse the inequality sign, and students will have opportunities to practice with example problems and a quiz.

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Solving Two-Step Inequalities and Graphing Solutions

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  1. 7 13 1 16 16 8 x = – Week 23, Day Three HW # 78 - p. 150 & 151 # 1-54 even Warm up Solve. Graph the solution. 1.6x + 36 = 2x 2. 4x – 13 = 15 + 5x 3. 5(x – 3) = 2x + 3 4. + x = x = –9 x = –28 x = 6 You need to do your audit TODAY!

  2. Warm Up Response ~$

  3. Homework Check p. 146 & 147 # 1-44 even

  4. Goals for Today • 3-8 Solving Two-Step Inequalities • Examples and Notes from p. 148 & 149 • CW: Inequalities Worksheet A

  5. Extra slides for practice

  6. When you solved two-step equations, you used the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities.

  7. 1 2 3 4 5 6 7 12 4x > 4 4 Additional Example 1A: Solving Two-Step Inequalities Solve and graph. 4x + 1 > 13 4x + 1 > 13 – 1– 1Since 1 is added to 4x, subtract 1 from both sides. 4x > 12 Since x is multiplied by 4, divide both sides by 4. x > 3

  8. Additional Example 1A Continued Check According to the graph 4 should be a solution and 2 should not be a solution. x > 3 x > 3 Substitute 4 for x. Substitute 2 for x. ? ? 4 > 3 2 > 3  So 4 is a solution. So 2 is not a solution.

  9. Remember! If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed.

  10. -6 -5 -4 -3 -2 -1 0 18 –9x –9 –9 Additional Example 1B: Solving Two-Step Inequalities Solve and graph. –9x + 7  25 –9x + 7  25 – 7– 7Subtract 7 from both sides. –9x 18 Divide each side by –9; change  to . x–2

  11. 1 2 3 4 5 6 7 10 5x > 5 5 Check It Out! Example 1A Solve and graph. 5x + 2 > 12 5x + 2 > 12 – 2– 2Subtract 2 from both sides. 5x > 10 Divide both sides by 5. x > 2

  12. Check It Out! Example 1A Continued Check According to the graph 4 should be a solution and 1 should not be a solution. x > 2 x > 2 Substitute 4 for x. Substitute 1 for x. ? ? 4 > 2 1 > 2  So 4 is a solution. So 1 is not a solution.

  13. -6 -5 -4 -3 -2 -1 0 16 –4x –4 –4 Check It Out! Example 1B –4x + 2  18 –4x + 2  18 – 2– 2Subtract 2 from both sides. –4x 16 Divide each side by –4; change  to . x–4

  14. 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? In order for the Spanish club to make a profit, the revenue must be greater than the cost. 1.25x > 55 + 0.15x

  15. 55 > 1.10x 1.10 1.10 Additional Example 3 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.

  16. Check It Out! Example 3 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? In order for the French club to make a profit, the revenue must be greater than the cost. 2.5x > 45 + 0.25x

  17. 45 > 2.25x 2.25 2.25 Check It Out! Example 3 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 French club must sell more than 20 bumper stickers to make a profit.

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

  19. 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

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