1. 11-4 Inequalities Course 2 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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 Course 2 11-4 Inequalities k = 1212

3. 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 cm 2. What is the area of each rectangle? 56 cm 2 and 14 cm 2 Course 2 11-4 Inequalities

4. Learn to read and write inequalities and graph them on a number line. Course 2 11-4 Inequalities

5. Vocabulary inequality algebraic inequality solution set compound inequality Insert Lesson Title Here Course 2 11-4 Inequalities

6. An inequality states that two quantities either are not equal or may not be equal. An inequality uses one of the following symbols: SymbolMeaningWord Phrases < > ≤ ≥ is less than is greater than is greater than or equal to is less than or equal to Fewer than, below More than, above At most, no more than At least, no less than Course 2 11-4 Inequalities

7. Write an inequality for each situation. Additional Example 1: Writing Inequalities A. There are at least 15 people in the waiting room. number of people ≥ 15 B. The tram attendant will allow no more than 60 people on the tram. number of people ≤ 60 “At least” means greater than or equal to. “No more than” means less than or equal to. Course 2 11-4 Inequalities

8. Write an inequality for each situation. Try This: Example 1 A. There are at most 10 gallons of gas in the tank. gallons of gas ≤ 10 B. There is at least 10 yards of fabric left. yards of fabric ≥ 10 “At most” means less than or equal to. “At least” means greater than or equal to. Course 2 11-4 Inequalities

9. 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. Course 2 11-4 Inequalities

10. 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 Course 2 11-4 Inequalities

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

12. Graph each inequality. Try This: Example 2A & 2B –3 –2 –1 0 1 2 3 A. p ≤ 2 2 is a solution, so draw a closed circle at 2. Shade the line to the left of 2. B. e > –2 –3 –2 –1 0 1 2 3 –2 is not a solution, so draw an open circle at –2. Shade the line to the right of –2. Course 2 11-4 Inequalities

13. A compound inequality is 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. The compound inequality –2 < y and y < 4 can be written as –2 < y < 4. Writing Math Course 2 11-4 Inequalities

14. Graph each compound inequality. Additional Example 3A: Graphing Compound Inequalities 0 1 2 3 456 1 2 3 456 – – ––– – A. m ≤ –2 or m > 1 First graph each inequality separately. m ≤ –2m > 1 Then combine the graphs. 0 246 246 –– – 0 2 4 6 –2–4 –6 º The solutions of m ≤ –2 or m > 1 are the combined solutions of m ≤ –2 or m > 1. Course 2 11-4 Inequalities

15. Graph each compound inequality Additional Example 3B: Graphing Compound Inequalities B. –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 0 2 4 6 24 6 ––– 0 2 4 6 –2–4 –6 º Then combine the graphs. Remember that –3 < b ≤ 0 means that b is between –3 and 0, and includes 0. 0 1 2 3 456 1 2 3 456 – – ––– – Course 2 11-4 Inequalities

16. Graph each compound inequality. Try This: Example 3A 0 1 2 3 456 1 2 3 456 – – ––– – A. w < 2 or w ≥ 4 First graph each inequality separately. w < 2W ≥ 4 Then combine the graphs. 0 246 246 –– – 0 2 4 6 –2–4 –6 The solutions of w < 2 or w ≥ 4 are the combined solutions of w < 2 or w ≥ 4. Course 2 11-4 Inequalities

17. Graph each compound inequality Try This: Example 3B B. 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 0 2 4 6 24 6 ––– 0 2 4 6 –2–4 –6 º Then combine the graphs. Remember that 5 > g ≥ –3 means that g is between 5 and –3, and includes –3. 0 1 2 3 456 1 2 3 456 – – ––– – Course 2 11-4 Inequalities

18. Lesson Quiz: Part 1 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. number of eggs ≥ 12 people in the theater ≤ 220 Insert Lesson Title Here people attending the meeting < 14 Course 2 11-4 Inequalities

19. Lesson Quiz: Part 2 Graph the inequalities. 4. x > –1 Insert Lesson Title Here 0 º 135 123 – – – 5. x ≥ 4 or x < –1 0 º 135 135 – – – Course 2 11-4 Inequalities