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**LESSON 6-3 COMPOUND INEQUALITIES**

Objective: To solve and graph inequalities containing “and” or “or”.

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**Real-World Connection**

Do you like to swim? Does your family have a swimming pool? Determining how to keep the water in a pool chemically balanced is one way this skill is used in real-world situations.

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Vocabulary compound inequality – two inequalities joined by the word and or the word or solution for and inequalities – (Intersection) any number that makes both inequalities true solution for or inequalities – (Union) any number that makes either inequality true

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STEPS for AND problems Write the compound inequality as two inequalities joined by and. Solve each inequality. Simplify and write the solutions as one statement. Graph the solution.

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**Translate the verbal phrase into an inequality**

Translate the verbal phrase into an inequality. Then graph the inequality. a. All real numbers that are greater than – 2 and less than 3. Inequality: – 2 < x < 3 Graph: b. All real numbers that are less than 0 or greater than or equal to 2. x < 0 or x ≥ 2 Inequality: Graph:

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**All real numbers that are less than –1 or greater than or equal to 4.**

c. Inequality: x < –1 or x ≥ 4 d. All real numbers that are greater than or equal To –3 and less than 5. Inequality: x ≥ –3 and x < 5 = –3 ≤ x < 5

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**Write and graph a real-world inequality:**

CAMERA CARS A crane sits on top of a camera car and faces toward the front. The crane’s maximum height and minimum height above the ground are shown. Write and graph a compound inequality that describes the possible heights of the crane.

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**Write and graph a real-world inequality:**

SOLUTION Let h represent the height (in feet) of the crane. All possible heights are greater than or equal to 4 feet and less than or equal to 18 feet. So, the inequality is 4 ≤ h ≤ 18.

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**You try… Solve each inequality and graph the solution.**

1) -6 < 3x < 15 2) -3 < 2x-1 < 7

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You try… 3) 7 < -3a + 1 < 13 Solution: -4 < n < -2

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**You try… 4) The books were priced between $3.50 and $6.00, inclusive.**

Solution: < b < 6

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**To solve OR problems, solve each inequality separately.**

5) Solve –2x + 7 > 3 or 3x – 4 > 5.

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You try… 6) Write an inequality that represents all real numbers that are at most –5 or at least 3. Graph your solution.

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**Remember, when written like this, it is an AND problem! **

7) Graph 3 < 2m – 1 < 9 Remember, when written like this, it is an AND problem! 3 < 2m – 1 AND 2m – 1 < 9 Solve each inequality. Graph the intersection of 2 < m and m < 5. 5 -

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**Which inequalities describe the following graph?**

-2 -1 -3 o y > -3 or y < -1 y > -3 and y < -1 y ≤ -3 or y ≥ -1 y ≥ -3 and y ≤ -1 Answer Now

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**Which is equivalent to -3 < y < 5?**

y > -3 or y < 5 y > -3 and y < 5 y < -3 or y > 5 y < -3 and y > 5 Answer Now

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**Which is equivalent to x > -5 and x ≤ 1?**

Answer Now

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**Summary Re-write the compound inequality into two problems and solve.**

For and problems, combine the solutions into one statement. For or problems, solve each separately. Graph.

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Homework: Study for quiz Wednesday, 1/15 Quiz covers chapter 6 sections 1-3

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**Explain the difference between “and” vs. “or” compound inequalities.**

Exit Ticket Explain the difference between “and” vs. “or” compound inequalities. How do you solve a compound inequality?

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