LESSON 6-3 COMPOUND INEQUALITIES

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

Real-World Connection   Does your family have a swimming pool?

 Do you like to swim?

Determining how to keep the water in a pool chemically balanced is one way this skill is used in real-world situations.

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

STEPS for AND problems 1.

2.

3.

4.

Write the compound inequality as two inequalities joined by and.

Solve each inequality.

Simplify and write the solutions as one statement.

Graph the solution.

a .

Translate the verbal phrase into an inequality. Then graph the inequality.

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.

Inequality :

x

< 0

or x

≥ 2

Graph :

c.

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

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

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.

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.

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

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

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

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

 Solution: 3.50 < b < 6

To solve OR problems, solve each inequality separately.

5) Solve –2x + 7 > 3 or 3x – 4 > 5.

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

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 0 5

Which inequalities describe the following graph?

o

-3 -2

o

-1

1.

y > -3 or y < -1 2.

3.

4.

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

Which is equivalent to -3 < y < 5?

1.

2.

3.

4.

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

1.

2.

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

-5 < x ≤ 1 -5 > x ≥ 1 3.

4.

-5 > x ≤ 1 -5 < x ≥ 1 Answer Now

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.

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

Exit Ticket  Explain the difference between “and” vs. “or” compound inequalities.

 How do you solve a compound inequality?