1. CONFIDENTIAL 1 Algebra1 Graphing and Writing Inequalities

2. CONFIDENTIAL 2 Warm Up 1) 2b - 6 = b + 3 3) 2 (y + 1) = 2y + 1 4) x + (x + 1) + (x + 2) 5) 5 + (x + 3) + 5 + 2 (x + 3) Solve each equation. 2) -3 (2 - x) = 5x + 2 Simplify each expression. 1) b = 9 2) b = 9 3) contradictory statement 4) 3x + 3 5) 3x + 19

3. CONFIDENTIAL 3 Inequalities An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs: A < B A > B A ≤ B A ≥ B A ≠ B Where, A and B are some integer value.

4. CONFIDENTIAL 4 Any number that makes an inequality true is a solution of the inequality. For example: -4 is a solution of y ≥ -5 because -4 ≥ -5. Inequalities, like equations, can be true, false or open. 15 > 11 6 < 3 a < 20 This sentence is true. This sentence is false. This sentence is open. It is neither true nor false until “a" replaced with a number.

5. CONFIDENTIAL 5 <>≤≥≠  Less than  Fewer than  Greater than  More than  Exceeds  Less than or equal to  No more than  At most  Greater than or equal to  No less than  At least  A is not equal to B. Many situations in real life can be described using inequalities. The table below shows some of the common phrases and corresponding inequalities.

6. CONFIDENTIAL 6 Describe the solutions of 3 + x < 9 in words. Identifying Solutions of Inequalities Test values of x that are positive, negative, and 0. When the value of x is a number less than 6, the value of 3 + x is less than 9. When the value of x is 6, the value of 3 + x is equal to 9. When the value of x is a number greater than 6, the value of 3 + x is greater than 9. x-2.7505.9966.01 3 + x-5.7538.9999.01 3 + x < 9-2.75 < 93 < 98.99 < 99<99.01< 9 Solution?Yes No

7. CONFIDENTIAL 7 1. Describe the solutions of 2p > 8 in words. Now you try! 1) p > 4. When the value of p is a number greater than 4, the value of 2p > 8 is greater than 8.

8. CONFIDENTIAL 8 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. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 6 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 that an endpoint is not a solution, draw an empty circle.

9. CONFIDENTIAL 9 Graphing Inequalities WORDS ALGEBRA All real numbers less than 5 x < 5 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 5 All real numbers greater than -1 x > -1 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x > -1

10. CONFIDENTIAL 10 01 2 34 5 678 9 11 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x ≥ 0 WORDS ALGEBRA All real numbers greater than or equal to 0 x ≥ 0 All real numbers less than or equal to 1 x ≤ 1 2 2 0 1 2 3 4 5 6 -5 -4 -3 -2 x ≥ 1 2

11. CONFIDENTIAL 11 Graphing Inequalities The solution of inequality x < 9 represented on the number line is 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 9 A) Solve: Graph the solution of he inequality, x < 9 on the number line. An open dot shows that 9 is not a solution. Draw an arrow pointing to the left. Shade all points to the left of 9.

12. CONFIDENTIAL 12 The solution of inequality x ≥ -3 represented on the number line is 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x ≥ -3 B) Solve: Graph the solution of the inequality, x ≥ -3 on the number line. An solid circle shows that -3 is a solution. Draw an arrow pointing to the right. Shade all points to the right of -3.

13. CONFIDENTIAL 13 Graph each inequality. Now you try! 1) c > 2.52) 2 2 - 4 ≥ w 2.5 0 1 2 3 4 5 6 -5 -4 -3 -2 2.5 1) 2) 2.5 0 1 2 3 4 5 6 -5 -4 -3 -2 0

14. CONFIDENTIAL 14 Writing an Inequality from a Graph Write the inequality shown by each graph. 2.5 0 1 2 3 4 5 6 -5 -4 -3 -2 Use any variable. The arrow points to the right, so use either > or ≥. The empty circle at 4.5 means that 4.5 is not a solution, so use >. x > 2.5 A) 2.5 x > 2.5

15. CONFIDENTIAL 15 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 B) Use any variable. The arrow points to the left, so use either < or ≤. The solid circle at -5 means that -5 is a solution, so use ≤. x ≤ -5 - 5

16. CONFIDENTIAL 16 1. Write the inequality shown by the graph. Now you try! 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 - 0.5 The open dot and the left arrow from point -0.5 shows the value x < -0.5

17. CONFIDENTIAL 17 Solution: A) The members of a lightweight crew team can weigh no more than 165 pounds each. Define a variable and write an inequality for the acceptable weights of the team members. Graph the solutions. Sports Application Let w represent the weights that are allowed. i.e., w ≤ 165 Stop the graph at 0 because a person’s weight must be a positive number. 105 120 135 150 165 01530 45 60 75 90

18. CONFIDENTIAL 18 1. A store’s employees earn at least $8.25 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions. Now you try! 1) x ≥ 8.25 2 3 4 5 6 7 8 9 10 0 1 8.25

19. CONFIDENTIAL 19 Assessment Describe the solutions of each inequality in words. 1) g - 5 ≥ 62) -2 < h + 1 3) 20 > 5t 1) g ≥ 11. When the value of g is a number greater than 11, the value of g - 5 is greater than 9. 2) h > -3. When the value of h is a number greater than -3, the value of h + 1 is greater than -2. 3) t < 4. When the value of t is a number less than 4, the value of 5t is less than 20.

20. CONFIDENTIAL 20 Graph each inequality. 4) x < -5 5) (4 - 2) 3 > m 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 4) 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 5) -5 8

21. CONFIDENTIAL 21 Write the inequality shown by each graph. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 6) x ≤ - 5 6) 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 7) x > 2.5 7)

22. CONFIDENTIAL 22 Define a variable and write an inequality for each situation. Graph the solutions. 8) There must be at least 20 club members present in order to hold a meeting. 9) A trainer advises an athlete to keep his heart rate under 140 beats per minute. 10) The maximum speed allowed on Main Street is 25 miles per hour. 8) x ≥ 20 9) x < 140 10) x ≤ 25

23. CONFIDENTIAL 23 Inequalities An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs: A < B A > B A ≤ B A ≥ B A ≠ B Where, A and B are some integer value. Let’s review

24. CONFIDENTIAL 24 <>≤≥≠  Less than  Fewer than  Greater than  More than  Exceeds  Less than or equal to  No more than  At most  Greater than or equal to  No less than  At least  A is not equal to B. Many situations in real life can be described using inequalities. The table below shows some of the common phrases and corresponding inequalities.

25. CONFIDENTIAL 25 Describe the solutions of 3 + x < 9 in words. Identifying Solutions of Inequalities Test values of x that are positive, negative, and 0. When the value of x is a number less than 6, the value of 3 + x is less than 9. When the value of x is 6, the value of 3 + x is equal to 9. When the value of x is a number greater than 6, the value of 3 + x is greater than 9. x-2.7505.9966.01 3 + x-5.7538.9999.01 3 + x < 9-2.75 < 93 < 98.99 < 99<99.01< 9 Solution?Yes No

26. CONFIDENTIAL 26 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. 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 6 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 that an endpoint is not a solution, draw an empty circle.

27. CONFIDENTIAL 27 Graphing Inequalities The solution of inequality x < 9 represented on the number line is 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x < 9 A) Solve: Graph the solution of he inequality, x < 9 on the number line. An open dot shows that 9 is not a solution. Draw an arrow pointing to the left. Shade all points to the left of 9.

28. CONFIDENTIAL 28 The solution of inequality x ≥ -3 represented on the number line is 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 x ≥ -3 B) Solve: Graph the solution of the inequality, x ≥ -3 on the number line. An solid circle shows that -3 is a solution. Draw an arrow pointing to the right. Shade all points to the right of -3.

29. CONFIDENTIAL 29 Writing an Inequality from a Graph Write the inequality shown by each graph. 2.5 0 1 2 3 4 5 6 -5 -4 -3 -2 Use any variable. The arrow points to the right, so use either > or ≥. The empty circle at 4.5 means that 4.5 is not a solution, so use >. x > 2.5 A) 2.5 x > 2.5

30. CONFIDENTIAL 30 01 2 34 5 678 9 1011 12 -11 -10 -9-8 -7-6-5 -4 -3-2 B) Use any variable. The arrow points to the left, so use either < or ≤. The solid circle at -5 means that -5 is a solution, so use ≤. x ≤ -5 - 5

31. CONFIDENTIAL 31 You did a great job today!