Inequalities

Inequality

100

x/11 >= 10

x >= 110

[110, infinity)

200

-7n - 10 >= 60

n <= -10

(-infinity, -10]

300

-(k - 2) > - k - 20

True Statement

All real number (-infinity, infinity)

400

-36 + 6x > -8(x + 2) + 4x

x > 2

(2, infinity)

500

- (4 - 5p) + 3 >= -2(8 - 5p)

p <= 3

(- infinity, 3]

OR Type

100

6m >= -24

OR

m - 7 < - 12

m >= -4 OR m < -5 

(- infinity, -5) U [-4, infinity)

200

3 + 7r > 59

OR

-6r - 3 > 33

r > 8 OR r <-6

(-infinity, -6) U (8, infinity)

300

1 + 5k <= 7k - 3

OR

k - 10 > 2k + 10

k >= 2 OR k < -20

(-infinity, -20) U [2, infinity)

400

8 - 10r <= 8 + 4r

OR

-6 + 8r < 2 + 8r

True Statements

All Real Numbers (-infinity, infinity)

500

Each type of fish thrives in a specific range of temperatures. The optimum temperatures for sharks range from 18 degrees C to 22 degrees C. Write an inequality that represents the temperature where sharks will NOT thrive.

x < 18 OR x > 22 [degrees C]

AND Type

100

- 8 + b < -3 

AND

4b < 20

b < 5 for both

(-infinity, 5)

200

-6n <= 12 

AND

n/3 <= 2

n >= -2 AND n <= 6 

(AKA -2 <= n <= 6)

[-2, 6]

300

-4 < 8 - 3m <= 11

-1 <= m < 4

[-1, 4)

400

-16 <= 2n - 10 <= -22

No Solution

False Statement

500

2x + 9 >= 10x + 1
AND

3x - 2 < 7x + 2

-1 < x <= 1

(-1, 1]

Compound Inequality

100

r + 5 >= 12

OR

r/9 < 0

r >= 7 OR r < 0

(-infinity, 0) U [7, infinity)

200

Nine less than a number is NO MORE than 8 and is NO LESS than 3

(Double points to simplify)

3 <= x - 9 <= 8

Simplified: 12 <= x <= 17

300

4n + 8 < 3n - 6

OR

10n - 8 >= 9 + 9n

n < -14 OR n >=17

(-infinity, -14) U [17, infinity)

400

One half of a number is increased by 3, is greater than zero or less than or equal to negative 3

0.5x + 3 > 0

OR

0.5x + 3 <= -3

500

About 20% of the time you sleep is spend in rapid eye movement (REM) sleep, which is associated with dreaming. If an adult sleeps between 7 and 8 hours, how much time is spent in REM sleep?

7 < 5h < 8

Simplified: 1.4 < h < 1.6

Absolute Inequality

100

|x| <= 8

[-8, 8]

200

|x + 3| < 4

-4 < x + 3 < 4

-7 < x < 1

(-7, 1)

300

|2x - 4| > 6

 2x - 4 < -6 OR 2x - 4 > 6

x < -1 OR x > 5

(-infinity, -1) U (5, infinity)

400

-2 + 3|5 - x| <= 4

3|5 - x| <= 6

|5 - x| <= 2

-2 <= 5 - x <= 2

-7 <= -x <= -3

7>= x >= 3

[3,7]

500

4 - 6|-6 - 3x| <= -5

- 6|-6 - 3x| <= -9

|-6 - 3x| >= 9/6 (or 3/2 = 1-1/2)

-3/2 >= -6 - 3x    OR   3/2 <= -6 - 3x

9/2 >= - 3x      OR     15/2 <= -3x

-9/6 <= x       OR     -15/6 >= x

(-infinity, -5/2] U [-3/2, infinity)