Algebra

6.3 Solving Compound Inequalities

The difference between the words “and” and “or”

If your mother tells you to do something, do the words she uses make a difference?

“Please take out the garbage and make your bed.”“Please take out the garbage or make your bed.”

How many chores do you have if she uses the word “and”?

How many chores do you have if she uses the word “or”?

Two. You have to do both chores.

One. You only have to do one chore or the other.

Compound Inequality (2 types)

Two inequalities connected by and or or.And

-2 < x ≤ 3 means -2 < x and x ≤ 3All real numbers greater than -2 and less than or equal to 3.

Orx ≤ -4 or x > 2All real numbers less than or equal to -4 or greater than 2.

Both inequalities must be true

Only one inequality must be true

Graphs of Compound Inequalities

And

Look like this:

OrLook like this:

Graphs of Compound Inequalities

And-2 < x ≤ 3 means -2 < x and x ≤ 3

All real numbers greater than -2 and less than or equal to 3.

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Graphs of Compound Inequalities

Or

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

x ≤ -4 or x > 2

All real numbers less than or equal to -4 or greater than 2.

Example

Solve:-5 ≤ 2x +3 < 7

-3 - 3 -3

-8 ≤ 2x < 4

2 2 2

-4 ≤ x < 2

-5 -4 -3 -2 -1 0 1 2 3-5 -4 -3 -2 -1 0 1 2 3

Example

Solve:-2 < -2 - x < 1+2 +2 +2

0 < - x < 3

-1 -1 -1 0 > x >

-3

-5 -4 -3 -2 -1 0 1 2 3-5 -4 -3 -2 -1 0 1 2 3

-3 < x

< 0

Rewrite innumber line order

You try

Solve:-5 ≤ -2x +3 < 9-3 - 3 -3

-8 ≤ -2x < 6

-2 -2 -24 ≥ x > -3

-4 -3 -2 -1 0 1 2 3 4 5-4 -3 -2 -1 0 1 2 3 4 5

-3 < x ≤ 4 Number line order

Example

Solve:6x – 5 < 7 or 8x + 6 ≥ 46

+5 +5 or -6 -6

x < 2 or x ≥ 5

-1 0 1 2 3 4 5 6 7-1 0 1 2 3 4 5 6 7

6x < 12 or 8x ≥ 40

6 6 8 8

You try

Solve:3x + 1 ≤ 4 or 2x - 5 > 7

-1 -1 or +5 +5

x ≤ 1 or x > 6

-1 0 1 2 3 4 5 6 7-1 0 1 2 3 4 5 6 7

3x ≤ 3 or 2x > 12

3 3 2 2

A few together from the HW

P. 349 #18, #26, #32

Homework

Pg. 349 #18-35, 55-69