Warm Up Compare. Use < or >. 1. 5 7 2. –3 –4 1. 5 7 2. –3 –4 3. 2.5 –2.7 4. –8 –7 Solve. 5. 4 + y = 16 6. m – 7 = 14 7. –3 = 8 + w 8. 7 = t + 10 < > > < 12 21 –11 –3

When you add or subtract the same number on both sides of an inequality, the resulting statement will still be true. –2 < 5 +7 +7 5 < 12 You can find solution sets of inequalities the same way you find solutions of equations, by isolating the variable.

Additional Example 1A: Solving Inequalities by Adding or Subtracting Solve and graph the inequality. x + 3 > –5 x + 3 > –5 Since 3 is added to x, subtract 3 from both sides. –3 –3 x > –8 –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

Additional Example 1A Continued Check According to the graph –4 should be a solution and –9 should not be a solution. x + 3 > –5 x + 3 > –5 ? Substitute –4 for x. ? Substitute –9 for x. –4 + 3 > –5 –9 + 3 > –5 ? ? –1 > –5 –6 > –5  So –4 is a solution. So –9 is not a solution.

Additional Example 1B: Solving Inequalities by Adding or Subtracting Solve and graph the inequality. m – 4 ≥ –2 m – 4 ≥ –2 Since 4 is subtracted from m, add 4 to both sides. + 4 + 4 m ≥ 2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12

Additional Example 1C: Solving Inequalities by Adding or Subtracting Solve and graph the inequality. r + 3 ≤ –3 r + 3 ≤ –3 Since 3 is added to r, subtract 3 from both sides. – 3 –3 r ≤ –6 –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

Additional Example 1D: Solving Inequalities by Adding or Subtracting Solve and graph the inequality. 34 14 5 > n + 1 34 14 5 > n + 1 Since 1¼ is added to n, subtract 1¼ from both sides. 14 14 – 1 – 1 12 4 > n –7 -6 –5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Solve and graph the inequality. Check It Out! Example 1A Solve and graph the inequality. x + 4 > –2 x + 4 > –2 Since 4 is added x, subtract 4 from both sides. –4 –4 x > –6 Hiding Slide 10 –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

Check It Out! Example 1A Continued According to the graph 2 should be a solution and –8 should not be a solution. x + 4 > –2 x + 4 > –2 ? Substitute 2 for x. ? Substitute –8 for x. 2 + 4 > –2 –8 + 4 > –2 ? ? 6 > –2 –4 > –2  Hiding Slide 11 So 2 is a solution. So –8 is not a solution.

Solve and graph the inequality. Check It Out! Example 1B Solve and graph the inequality. w – 8 ≥ –3 w – 8 ≥ –3 Since 8 is subtracted from w, add 8 to both sides. + 8 + 8 w ≥ 5 Hiding Slide 12 –1 0 1 2 3 4 5 6 7 8 9 10 11 12

Solve and graph the inequality. Check It Out! Example 1C Solve and graph the inequality. c + 6 ≤ –1 c + 6 ≤ –1 Since 6 is added to c, subtract 6 from both sides. – 6 – 6 c ≤ –7 Hiding Slide 13 –9 -8 –7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

Solve and graph the inequality. Check It Out! Example 1D Solve and graph the inequality. 23 13 3 > n + 1 23 13 3 > n + 1 13 13 Since 1 is added to n, subtract 1 from both sides. 13 – 1 – 1 13 13 2 > n Hiding Slide 14 –7 -6 –5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Additional Example 2: Sports Application While training for a race, Ann’s goal is to run at least 3.5 miles each day. She has already run 1.8 miles today. Write and solve an inequality to find out how many more miles she must run today. Let m = the number of additional miles. 1.8 miles plus additional miles is at least 3.5 miles. 1.8 + m ≥ 3.5 1.8 + m ≥ 3.5 –1.8 –1.8 Since 1.8 is added to m, subtract 1.8 from both sides. m ≥ 1.7 Ann should run at least 1.7 more miles.

Additional Example 2 Continued Check 1.8 + m ≥ 3.5 1.8 + 2 ≥ 3.5 ? 2 is greater than 1.7. Substitute 2 for m. 3.8 ≥ 3.5 ? 1.8 + m ≥ 3.5 1 is less than 1.7. Substitute 1 for m. 1.8 + 1 ≥ 3.5 ? 2.8 ≥ 3.5 ? x

Check It Out! Example 2 Tim’s company produces recycled paper. They produce 60.5 lb of paper each day. They have already produced at least 20.2 lb today. Write and solve an inequality to find out how many more pounds Tim’s company must produce. Let p = the number of additional pounds of paper. 20.2 lbs plus additional pounds is at least 60.5 lb. 20.2 + p ≥ 60.5 20.2 + p ≥ 60.5 Hiding Slide 17 Since 20.2 is added to p, subtract 20.2 from both sides. –20.2 – 20.2 p ≥ 40.3 Tim’s company should produce at least 40.3 lb more of paper.

Check It Out! Example 2 Continued 20.2 + 41 ≥ 60.5 ? 41 is greater than 40.3. Substitute 41 for p. 61.2 ≥ 60.5 ? 20.2 + p ≥ 60.5 Hiding Slide 18 40 is less than 40.3. Substitute 40 for p. 20.2 + 40 ≥ 60.5 ? 60.2 ≥ 60.5 ? x

• • Lesson Review: Part I Solve and graph each inequality. 3. –5.1 ≤ x – 5.1 g < 4 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 • s ≥ –1 -4 -3 -2 -1 0 1 2 • x ≥ 0 y < 1/5 2/5 3/5 4/5 1 1 1/5 45 15 4. 3 + y > 4

Lesson Review: Part II 5. Tasha is folding letters for a fundraiser. She knows there are at least 300 letters, and she has already folded 125 of them. Write and solve an inequality to show how many more letters she must fold. 125 + x ≥ 300; x ≥ 175