Mastering Inequalities: Learn, Solve, Graph

1. Inequalities 12-4 Course 2 Warm Up Problem of the Day Lesson Presentation

2. Inequalities 12-4 Course 2 Warm Up Solve. 1. –21z + 12 = –27z 2. –12n – 18 = –6n 3. 12y – 56 = 8y 4. –36k + 9 = –18k z = –2 n = –3 y = 14 1 2 k =

3. Inequalities 12-4 Course 2 Problem of the Day The dimensions of one rectangle are twice as large as the dimensions of another rectangle. The difference in area is 42 cm2. What is the area of each rectangle? 56 cm2 and 14 cm2

4. Inequalities 12-4 Course 2 Learn to read and write inequalities and graph them on a number line.

5. Inequalities 12-4 Course 2 Insert Lesson Title Here Vocabulary inequality algebraic inequality solution set compound inequality

6. Inequalities 12-4 Course 2 An inequalitystates that two quantities either are not equal or may not be equal. An inequality uses one of the following symbols: is less than Fewer than, below is greater than More than, above is less than or equal to At most, no more than is greater than or equal to At least, no less than

7. Inequalities 12-4 Course 2 Additional Example 1: Writing Inequalities Write an inequality for each situation. A. There are at least 15 people in the waiting room. “At least” means greater than or equal to. number of people ≥ 15 B. The tram attendant will allow no more than 60 people on the tram. “No more than” means less than or equal to. number of people ≤ 60

8. Inequalities 12-4 Course 2 Check It Out: Example 1 Write an inequality for each situation. A. There are at most 10 gallons of gas in the tank. “At most” means less than or equal to. gallons of gas ≤ 10 B. There is at least 10 yards of fabric left. “At least” means greater than or equal to. yards of fabric ≥ 10

9. Inequalities 12-4 Course 2 An inequality that contains a variable is an algebraic inequality. A value of the variable that makes the inequality true is a solution of the inequality. An inequality may have more than one solution. Together, all of the solutions are called the solution set. You can graph the solutions of an inequality on a number line. If the variable is “greater than” or “less than” a number, then that number is indicated with an open circle.

10. Inequalities 12-4 Course 2 This open circle shows that 5 is not a solution. a > 5 If the variable is “greater than or equal to” or “less than or equal to” a number, that number is indicated with a closed circle. This closed circle shows that 3 is a solution. b ≤ 3

11. Inequalities 12-4 Course 2 Additional Example 2: Graphing Simple Inequalities Graph each inequality. A. n < 3 3 is not a solution, so draw an open circle at 3. Shade the line to the left of 3. –3 –2 –1 0 1 2 3 B. a ≥ –4 –4 is a solution, so draw a closed circle at –4. Shade the line to the right of –4. –6 –4 –2 0 2 4 6

12. Inequalities 12-4 Course 2 Check It Out: Example 2 Graph each inequality. A. p ≤ 2 2 is a solution, so draw a closed circle at 2. Shade the line to the left of 2. –3 –2 –1 0 1 2 3 B. e > –2 –2 is not a solution, so draw an open circle at –2. Shade the line to the right of –2. –3 –2 –1 0 1 2 3

13. Inequalities 12-4 Writing Math The compound inequality –2 < y and y < 4 can be written as –2 < y < 4. Course 2 A compound inequalityis the result of combining two inequalities. The words and and or are used to describe how the two parts are related. x > 3 or x < –1 –2 < y and y < 4 x is either greater than 3 or less than–1. y is both greater than –2 and less than 4. y is between –2 and 4.

14. Inequalities 12-4 –6 6 4 –4 –2 2 – – 2 4 6 0 0 – 6 4 2 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Course 2 Additional Example 3A: Graphing Compound Inequalities Graph each compound inequality. m ≤ –2 or m > 1 First graph each inequality separately. m ≤ –2 m > 1 • º Then combine the graphs. The solutions of m ≤ –2 or m > 1 are the combined solutions of m ≤ –2 or m > 1.

15. Inequalities 12-4 –6 6 4 –4 –2 2 0 6 6 – – – 4 4 2 2 0 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Course 2 Additional Example 3B: Graphing Compound Inequalities Graph each compound inequality –3 < b ≤ 0 –3 < b ≤ 0 can be written as the inequalities –3 < b and b ≤ 0. Graph each inequality separately. –3 < b b ≤ 0 • º Then combine the graphs. Remember that –3 < b ≤ 0 means that b is between –3 and 0, and includes 0.

16. Inequalities 12-4 Reading Math -3 < b is the same as b > -3. Course 2

17. Inequalities 12-4 –6 6 4 –4 –2 2 – – 2 4 6 0 0 – 6 4 2 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Course 2 Check It Out: Example 3A Graph each compound inequality. w < 2 or w ≥ 4 First graph each inequality separately. w < 2 W ≥ 4 Then combine the graphs. The solutions of w < 2 or w ≥ 4 are the combined solutions of w < 2 or w ≥ 4.

18. Inequalities 12-4 –6 6 4 –4 –2 2 0 6 6 – – – 4 4 2 2 0 – – – – – – 1 1 6 5 4 2 0 2 4 5 6 3 3 Course 2 Check It Out: Example 3B Graph each compound inequality 5 > g ≥ –3 5 > g ≥ –3 can be written as the inequalities 5 > g and g ≥ –3. Graph each inequality separately. 5 > g g ≥ –3 • º Then combine the graphs. Remember that 5 > g ≥ –3 means that g is between 5 and –3, and includes –3.

19. Inequalities 12-4 Course 2 Insert Lesson Title Here Lesson Quiz: Part I Write an inequality for each situation. 1. No more than 220 people are in the theater. 2. There are at least a dozen eggs left. 3. Fewer than 14 people attended the meeting. people in the theater ≤ 220 number of eggs ≥ 12 people attending the meeting < 14

20. Inequalities 12-4 º – 1 3 5 – 0 – 3 2 1 • º – 1 3 5 – 0 – 5 3 1 Course 2 Insert Lesson Title Here Lesson Quiz: Part II Graph the inequalities. 4.x > –1 5.x ≥ 4 or x < –1