3.6 Compound inequalities:

Inverse Operations: Operations that undo another operation.

Isolate: The use of inverse operations used to leave a variable by itself.

Compound Inequalities: two distinct inequalities joined by the word and or the word or.

GOAL:

These inequalities have the following graphs when the word and is being used:

Compound inequalities:

Set Notation: {x|3 ≤ x ≤ 7}

Interval: [3, 7]

Compound inequalities:

Set Notation: {x| x < -2 or x ≥1}Interval: (-∞,-2)U[1, ∞)

These inequalities have the following graphs when the word or is being used:

WRITING Compound inequalities:We must be able to transform English sentences into math equations:

EX:What compound inequality

represents the phrase:

“ All real numbers that are greater than – 2 and less than or equal to 6”

SOLUTION:“ All real numbers that are greater than – 2 and less than or equal to 6”

All real numbers greater than -2 X > -2

andAll real numbers less than or equal to 6

X ≤ 6-2 < x ≤ 6

Set Notation: {x| -2 < x ≤ 6} Interval: (-2, 6]

YOU TRY IT:

What compound inequality represents the phrase:

“ All real numbers that are less than 0 or greater than 5”

SOLUTION: “ All real numbers that are less than 0 or greater than 5”

All real numbers less than 0 X < 0

orAll real numbers greater than 5 X > 5

x< 0, x > 5

Set Notation: {x| x < 0, x > 5} Interval: (-∞,0) U (5, ∞)

SOLVING Compound Inequalities:To solve inequalities we follow inverse operations just like we do to isolate a variable:

EX:What are the solutions of: - 2 ≤ 2m – 4 < -1

SOLUTION:- 2 ≤ 2m – 4 < -1 Given (and)

4 - 2 ≤ 2m <-1 + 4 Inverse of subt.

2 ≤ 2m < 3 ≤ m <

Like terms

1 ≤ m < 1.5

Inverse of mult.

Interval: [1, 1.5)

YOU TRY IT:

What are the solutions

to 3t + 2 < -7 or -4t + 5 < 1

SOLUTION:3t + 2 < -7 -4t + 5 < 1

3t < -7 -2 -4t < 1 - 5

t < -3 -4t < -4

3t < -9

t > 1

or

Interval: (-∞,-3) U (1, ∞)

Real-World:You have taken a quiz and got

55%. You are about to take another quiz next week. If you

want to pass the portion of quizzes in the class you must get

an average between 70% and 79% What are the possible

percentages you must get on the next quiz?

Real-World: (SOLUTION)Quiz 1 = 55%

Quiz 2 = x%

Average

70% and 79%

70%

Thus in order for you to obtain a passing grade in your quizzes, you must get between 85% and 103% on your next quiz? (Good Luck!!!)

140

85

Real-World:A secondary 15-year old student should

consume no more than 2200 calories per day. A moderately active student should

consume between 2400 and 2800. An active 15-year-old student should consume between 2800 and 3200

calories per day. Model these ranges on a number line and represent them

in set and interval notations.

SOLUTION:A secondary 15-year old student

should consume no more than 2200 calories per day.

0 1000 2000

Set: { x | 0 < x ≤ 2200} Interval: (0, 2200]

2200

SOLUTION:A moderately active student should consume between 2400 and 2800

Calories per day.

0 1000 2000

Set: { x | 2400 < x ≤ 2800} Interval: (2400, 2800)

2800

3000

2400

SOLUTION:An active 15-year-old student should

consume between 2800 and 3200 calories per day.

0 1000 2000

Set: { x | 2800 < x ≤ 3200} Interval: (2800, 3200)

3200

3000

2800

VIDEOS: Compoundinequalities

https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_value_inequali/v/compund-inequalities

https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_value_inequali/v/compound-inequalities

VIDEOS: Compoundinequalities

https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_value_inequali/v/compound-inequalities-3

https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_value_inequali/v/compund-inequalities-2

CLASSWORK:

Page 204-206

Problems: As many as needed to master the

concept.