1. Copyright © Ed2Net Learning, Inc. 1 Solving Inequalities by Adding or Subtracting Solving Inequalities by Adding or Subtracting

2. Copyright © Ed2Net Learning, Inc. 2 Warm Up Translate the given statement into algebraic inequality. 1) A number exceeds 32. 2) At most 15 people applied. 3) Jenny scored at least 630 points. 4) A number less than 5.

3. Copyright © Ed2Net Learning, Inc. 3 Inequalities Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations. STEP1: Identify the variable. STEP2: To get the variable by itself, add the same number to or subtract the same number from each side of the inequality. STEP3: Check the solution. Solving inequality by Adding or Subtracting needs certain steps to be followed.

4. Copyright © Ed2Net Learning, Inc. 4 Addition property of inequality If you add the same amount to both sides of an inequality, the two sides will remain equal. In symbol: For all numbers a, b, and c: 1. If a > b, then a + c > b + c 2. If a < b, then a + c < b + c The rules are similar for a > b and a < b.

5. Copyright © Ed2Net Learning, Inc. 5 Subtraction property of inequality If you subtract the same amount from both sides of an inequality, the two sides will remain equal. In symbol: For all numbers a, b, and c: 1. If a > b, then a - c > b - c 2. If a < b, then a - c < b - c The rules are similar for a > b and a < b.

6. Copyright © Ed2Net Learning, Inc. 6 Using Addition and Subtraction to Solve Inequalities Solve each inequality and graph the solutions. A) x + 9 < 15 x + 9 < 15 -9 x + 0 < 6 x < 6 Since 9 is added to x, subtract 9 from both sides to undo the addition. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 6

7. Copyright © Ed2Net Learning, Inc. 7 B) d - 3 > -6 d - 3 > -6 +3 d + 0 > -3 d > -3 Since 3 is subtracted from d, add 3 to both sides to undo the subtraction. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x > -3

8. Copyright © Ed2Net Learning, Inc. 8 C) 0.7 ≥ n - 0.4 0.7 ≥ n - 0.4 + 0.4 1.1 ≥ n - 0 n ≤ 1.1 Since 0.4 is subtracted from n, add 0.4 to both sides to undo the subtraction. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 n ≤ 1.1

9. Copyright © Ed2Net Learning, Inc. 9 Solve each inequality and graph the solutions. Now you try! 1) s + 1 ≤ 10 2) 2 > -3 + t 1212

10. Copyright © Ed2Net Learning, Inc. 10 Since there can be an infinite number of solutions to an inequality, it is not possible to check all the solutions. You can check the endpoint and the direction of the inequality symbol. For example, the solutions of x + 9 < 15 are given by x < 6. Step 1 Check the endpoint. Substitute 6 for x in the related equation x + 9 = 15. The endpoint should be a solution of the equation. x + 9 = 15 6 + 9 15 15

11. Copyright © Ed2Net Learning, Inc. 11 Step 2 Check the inequality symbol. Substitute a number less than 6 for x in the original inequality. The number you choose should be a solution of the inequality. x + 9 < 15 4 + 9 < 15 13 < 15

12. Copyright © Ed2Net Learning, Inc. 12 -13 > x - 8 -13 + 8 > x – 8 + 8 -5 > x Add 8 to both sides Check: Try -5 and -7, numbers less than or equal to -5 -13 > -5 – 8 -13 > -7 – 8 -13 > -13 Both statements are true. -13 > -13 Solve: -13 > x - 8. Graph the solution.

13. Copyright © Ed2Net Learning, Inc. 13 x < -5 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 The solution is -5 > x. This means that the inequality is true for all numbers less than or equal to -5. The solution to x < 5 are graphed on the number line below. The point, 5, is included. Thus the circle is solid. The solution may also be written as x < -5.

14. Copyright © Ed2Net Learning, Inc. 14 Problem Solving Application The memory in Tania's camera phone allows her to take up to 20 pictures. Tania has already taken 16 pictures. Write, solve, and graph an inequality to show how many more pictures Tania could take. STEP1: Understand the Problem Tania can take up to, or at most, 20 pictures. Tania has taken 16 pictures already. STEP2: Make a Plan Let p represent the remaining number of pictures Tania can take. Number taken plus number remaining is at most 20 pictures. 16 + p ≤ 20

15. Copyright © Ed2Net Learning, Inc. 15 STEP3: Solve 16 + p ≤ 20 -16 -16 Since 16 is added to p, subtract 16 from both sides to undo the addition. It is not reasonable for Tania to take a negative or fractional number of pictures, so graph the nonnegative integers less than or equal to 4. Tania could take 0, 1, 2, 3, or 4 more pictures. p ≤ 4 STEP4: Solve Check: Check the endpoint, 4. Check a number less than 4. 16 + p = 20 16 + 4 20 20 16 + p ≤ 20 16 + 2 ≤ 20 18 ≤ 20

16. Copyright © Ed2Net Learning, Inc. 16 1) The Recommended Daily Allowance (RDA) of iron for a female in Sarah’s age group (14–18 years) is 15 mg per day. Sarah has consumed 11 mg of iron today. Write and solve an inequality to show how many more milligrams of iron Sarah can consume. Now you try!

17. Copyright © Ed2Net Learning, Inc. 17 Sports Application Josh can bench press 220 pounds. He wants to bench press at least 250 pounds. Write and solve an inequality to determine how many more pounds Josh must lift to reach his goal. Check your answer. Let p represent the number of additional pounds Josh must lift. SOLUTION: 220 pounds plus additional pounds is at least 250 pounds. 220 + p ≥ 250 -220 -220 Since 220 is added to p, subtract 220 from both sides to undo the addition. p ≥ 30

18. Copyright © Ed2Net Learning, Inc. 18 Check: Check the endpoint, 30. 220 + p = 250 220 + 30 250 250 220 + p ≥ 250 220 + 50 ≥ 250 270 ≥ 250 Check: Check a number greater than 30.

19. Copyright © Ed2Net Learning, Inc. 19 1) What If…? Josh wants to try to break the school record of 282 pounds. Write and solve an inequality to determine how many more pounds Josh needs to break the school record. Check your answer. Now you try!

20. Copyright © Ed2Net Learning, Inc. 20 Assessment Solve each inequality and graph the solutions. 1) 12 < p + 62) w + 3 ≥ 4 3) -5 + x ≤ -20 4) z - 2 > -11

21. Copyright © Ed2Net Learning, Inc. 21 Write an inequality for each sentence. Graph each inequality. 5) Roberto spent more than $10 for lunch. 6) Ben read no more than 26 pages. 7) The U.S. Constitution is more than 200 years old.

22. Copyright © Ed2Net Learning, Inc. 22 8) Jenny spent more than $6 at a fast food restaurant. If she had $25 to start, what was her remaining pocket money? 9) Christopher has $30 to buy a baseball cards for his collection. He has chosen a Greg Maddox rookie card that cost $8. What is the most he can spend on other cards? 10) On three extra credit problems on a math test, students are given up to 4 points each, provided that their scores do not exceed 100(points). What is the minimum raw scores students could have and still score 100 with extra credit?

23. Copyright © Ed2Net Learning, Inc. 23 Inequalities Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations. STEP1: Identify the variable. STEP2: To get the variable by itself, add the same number to or subtract the same number from each side of the inequality. STEP3: Check the solution. Solving inequality by Adding or Subtracting needs certain steps to be followed. Let’s review

24. Copyright © Ed2Net Learning, Inc. 24 Addition property of inequality If you add the same amount to both sides an inequality, the two sides will remain equal. In symbol: For all numbers a, b, and c: 1. If a > b, then a + c > b + c 2. If a < b, then a + c < b + c The rules are similar for a > b and a < b.

25. Copyright © Ed2Net Learning, Inc. 25 Subtraction property of inequality If you subtract the same amount from both sides an inequality, the two sides will remain equal. In symbol: For all numbers a, b, and c: 1. If a > b, then a - c > b - c 2. If a < b, then a - c < b - c The rules are similar for a > b and a < b.

26. Copyright © Ed2Net Learning, Inc. 26 Using Addition and Subtraction to Solve Inequalities Solve each inequality and graph the solutions. A) x + 9 < 15 x + 9 < 15 -9 x + 0 < 6 x < 6 Since 9 is added to x, subtract 9 from both sides to undo the addition. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 6

27. Copyright © Ed2Net Learning, Inc. 27 C) 0.7 ≥ n - 0.4 0.7 ≥ n - 0.4 + 0.4 1.1 ≥ n - 0 n ≤ 1.1 Since 0.4 is subtracted from n, add 0.4 to both sides to undo the subtraction. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 n ≤ 1.1

28. Copyright © Ed2Net Learning, Inc. 28 Problem Solving Application The memory in Tania's camera phone allows her to take up to 20 pictures. Tania has already taken 16 pictures. Write, solve, and graph an inequality to show how many more pictures Tania could take. STEP1: Understand the Problem Tania can take up to, or at most, 20 pictures. Tania has taken 16 pictures already. STEP2: Make a Plan Let p represent the remaining number of pictures Tania can take. Number taken plus number remaining is at most 20 pictures. 16 + p ≤ 20

29. Copyright © Ed2Net Learning, Inc. 29 STEP3: Solve 16 + p ≤ 20 -16 -16 Since 16 is added to p, subtract 16 from both sides to undo the addition. It is not reasonable for Tania to take a negative or fractional number of pictures, so graph the nonnegative integers less than or equal to 4. Tania could take 0, 1, 2, 3, or 4 more pictures. p ≤ 4 STEP4: Solve Check: Check the endpoint, 4. Check a number less than 4. 16 + p = 20 16 + 4 20 20 16 + p ≤ 20 16 + 2 ≤ 20 18 ≤ 20

30. Copyright © Ed2Net Learning, Inc. 30 You did a great job today!