1. Holt Algebra 1 3-1 Graphing and Writing Inequalities Warm Up Compare. Write, or =. 1. −3 2 3. 2. 6.5 6.3 < > > 4. 0.25= Tell whether the inequality x < 5 is true or false for the following values of x. 5. x = –10 T 6. x = 5 F 7. x = 4.99 T 8. x = T

2. Holt Algebra 1 3-1 Graphing and Writing Inequalities An inequality is a statement that two quantities are not equal. The quantities are compared by using the following signs: ≤ A ≤ B A is less than or equal to B. < A < BA < B A is less than B. > A > B A is greater than B. ≥ A ≥ B A is greater than or equal to B. ≠ A ≠ B A is not equal to B. A solution of an inequality is any value that makes the inequality true.

3. Holt Algebra 1 3-1 Graphing and Writing Inequalities An inequality like 3 + x < 9 has too many solutions to list. You can use a graph on a number line to show all the solutions. The solutions are shaded and an arrow shows that the solutions continue past those shown on the graph. To show that an endpoint is a solution, draw a solid circle at the number. To show an endpoint is not a solution, draw an empty circle.

4. Holt Algebra 1 3-1 Graphing and Writing Inequalities

5. Holt Algebra 1 3-1 Graphing and Writing Inequalities Graph each inequality. Check It Out! Example 1 a. c > 2.5 b. 2 2 – 4 ≥ w c. m ≤ –3

6. Holt Algebra 1 3-1 Graphing and Writing Inequalities Example 2: Writing an Inequality from a Graph Write the inequality shown by each graph.

7. Holt Algebra 1 3-1 Graphing and Writing Inequalities Write the inequality shown by the graph. Check It Out! Example 2

8. Holt Algebra 1 3-1 Graphing and Writing Inequalities Reading Math “No more than” means “less than or equal to.” “At least” means “greater than or equal to”.

9. Holt Algebra 1 3-1 Graphing and Writing Inequalities Example 3: Application Ray’s dad told him not to turn on the air conditioner unless the temperature is at least 85°F. Define a variable and write an inequality for the temperatures at which Ray can turn on the air conditioner. Graph the solutions.

10. Holt Algebra 1 3-1 Graphing and Writing Inequalities A store’s employees earn at least $8.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions. Check It Out! Example 3

11. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Solve one-step inequalities by using addition. Solve one-step inequalities by using subtraction. Objectives 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.

12. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Helpful Hint Use an inverse operation to “undo” the operation in an inequality. If the inequality contains addition, use subtraction to undo the addition.

13. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Example 1A: Using Addition and Subtraction to Solve Inequalities Solve the inequality and graph the solutions. x + 12 < 20

14. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting d – 5 > –7 Example 1B: Using Addition and Subtraction to Solve Inequalities Solve the inequality and graph the solutions.

15. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Example 1C: Using Addition and Subtraction to Solve Inequalities Solve the inequality and graph the solutions. 0.9 ≥ n – 0.3

16. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting a. s + 1 ≤ 10 Check It Out! Example 1 b. > –3 + t Solve each inequality and graph the solutions.

17. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting q – 3.5 < 7.5 Check It Out! Example 1c Solve the inequality and graph the solutions.

18. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting 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. The solutions of x + 9 < 15 are given by x < 6.

19. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Example 2: Problem-Solving Application Understand the problem 1 Sami has a gift card. She has already used $14 of the of the total value, which was $30. Write, solve, and graph an inequality to show how much more she can spend. The answer will be an inequality and a graph that show all the possible amounts of money that Sami can spend. List important information: Sami can spend up to, or at most $30. Sami has already spent $14.

20. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting 2 Make a Plan Example 2 Continued Write an inequality. Let g represent the remaining amount of money Sami can spend. g + 14 ≤ 30 Amount remaining plus $30. is at most amount used g + 14 ≤ 30

21. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Solve 3 Since 14 is added to g, subtract 14 from both sides to undo the addition. g + 14 ≤ 30 – 14 g + 0 ≤ 16 g ≤ 16 Draw a solid circle at 0 and16. Shade all numbers greater than 0 and less than 16. 0246810 12 14 16 18 10 Example 2 Continued

22. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Look Back4 Check Check the endpoint, 16. g + 14 = 30 16 + 14 30 30 Sami can spend from $0 to $16. Check a number less than 16. g + 14 ≤ 30 6 + 14 ≤ 30 20 ≤ 30 Example 2 Continued

23. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Check It Out! Example 2 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 without exceeding RDA.

24. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Check It Out! Example 2 Continued Understand the problem 1 The answer will be an inequality and a graph that show all the possible amounts of iron that Sami can consume to reach the RDA. List important information: The RDA of iron for Sarah is 15 mg. So far today she has consumed 11 mg.

25. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting 2 Make a Plan Write an inequality. Let x represent the amount of iron Sarah needs to consume. Amount taken plus 15 mg is at most amount needed 11 + x  15 11 + x  15 Check It Out! Example 2 Continued

26. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Solve 3 Since 11 is added to x, subtract 11 from both sides to undo the addition. 11 + x  15 x  4 Draw a solid circle at 4. Shade all numbers less than 4. 012345 6 7 8 9 10 x  4. Sarah can consume 4 mg or less of iron without exceeding the RDA. Check It Out! Example 2 Continued –11

27. Holt Algebra 1 3-2 Solving Inequalities by Adding or Subtracting Look Back4 Check Check the endpoint, 4. 11 + x = 15 11 + 4 15 15 Sarah can consume 4 mg or less of iron without exceeding the RDA. Check a number less than 4. 11 + 3  15 14  15 Check It Out! Example 2 Continued

28. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Solve one-step inequalities by using multiplication. Solve one-step inequalities by using division. Objectives Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number. The following rules show the properties of inequality for multiplying or dividing by a positive number. The rules for multiplying or dividing by a negative number appear later in this lesson.

29. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing

30. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Example 1A: Multiplying or Dividing by a Positive Number Solve the inequality and graph the solutions. 7x > –42

31. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Example 1B: Multiplying or Dividing by a Positive Number Solve the inequality and graph the solutions.

32. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Check It Out! Example 1a Solve the inequality and graph the solutions. 4k > 24

33. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing –50 ≥ 5q Check It Out! Example 1b Solve the inequality and graph the solutions.

34. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true.

35. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Caution! Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

36. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Example 2A: Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions. –12x > 84

37. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Example 2B: Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions.

38. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Check It Out! Example 2 Solve each inequality and graph the solutions. a. 10 ≥ –x b. 4.25 > –0.25h

39. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Example 3: Application $4.30 times number of tubes is at most $20.00. 4.30 p ≤ 20.00 Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy? Let p represent the number of tubes of paint that Jill can buy.

40. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing 4.30p ≤ 20.00 Example 3 Continued

41. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Check It Out! Example 3 A pitcher holds 128 ounces of juice. What are the possible numbers of 10-ounce servings that one pitcher can fill? 10 oz times number of servings is at most 128 oz 10 x ≤ 128 Let x represent the number of servings of juice the pitcher can contain.

42. Holt Algebra 1 3-3 Solving Inequalities by Multiplying or Dividing Check It Out! Example 3 Continued 10x ≤ 128