1. 11-4 Inequalities Warm Up Pick One to solve. (Or you can do both ) 1. –12n – 18 = –6n 2. 12y – 56 = 8y n = –3 y = 14

2. 11-4 Inequalities Today’s Goal: Learn to read and write inequalities and graph them on a number line.

3. 11-4 Inequalities Vocabulary inequality algebraic inequality solution set compound inequality

4. 11-4 Inequalities An inequality states that two quantities either are not equal or may not be equal. An inequality uses one of the following symbols: ≥ ≤ > < Word PhrasesMeaningSymbol 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

5. 11-4 Inequalities 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.

6. 11-4 Inequalities Write an inequality for each situation. Check It Out: 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.

7. 11-4 Inequalities 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.

8. 11-4 Inequalities 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

9. 11-4 Inequalities Graph each inequality. Additional Example 2: 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.

10. 11-4 Inequalities Graph each inequality. Check It Out: Example 2 –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.

11. 11-4 Inequalities 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

12. 11-4 Inequalities Graph each compound inequality. Additional Example 3A: Graphing Compound Inequalities 0 1 2 3 456 1 2 3 456 – – ––– – 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.

13. 11-4 Inequalities Graph each compound inequality Additional Example 3B: Graphing Compound Inequalities –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 – – ––– –

14. 11-4 Inequalities –3 –3. Reading Math

15. 11-4 Inequalities Graph each compound inequality. Check It Out: Example 3A 0 1 2 3 456 1 2 3 456 – – ––– – 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.

16. 11-4 Inequalities Graph each compound inequality Check It Out: Example 3B 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 – – ––– –