Solve |x-6|<= 7

means


After adding 6 to both sides we get

Solve |7-4x|>= |3|

so


has the same solution

Removing the absolute values leaves us with

1) 

or

2) 

1)


or
2)

Solve the inequality, if possible. Write the solution set in interval notation.

Solve the inequality |5x+11|>7 and write the solution set in interval form.

5x+11 > 7
5x > -4
x > -4/5

or
5x+11 < -7
5x < -18
x < -18/5
solution:
(-oo, -18/5)U(-4/5, oo)

Solve: |2x-1| < 9

|2x-1|<9
-9 < 2x-1 < 9
-9+1 < 2x-1+1 < 9+1
-8 < 2x < 10
-8/2 < 2x/2 < 10/2
-4 < x < 5

solve: -3+|2x+7| > 6

-3+|2x+7| > 6
-3+3+|2x+7| > 6+3
|2x+7| > 9
2x+7 > 9 or 2x+7 < -9
2x > 2 or 2x < -16
x > 1 or x < -8

Solve: -12<|-3x|+1

This is true for any real number because the right side of the inequality is always positive

3+|6x-1| < 14

|6x-1| < 11
-11 < 6x-1 < 11
-11+1 < 6x < 11+1
-10/6 < x < 2
-5/3 < x < 2

Write an absolute value equation or inequality representing all real numbers x at least 3 from 7

so we want:

x < 7-3 or x > 7+3
which means:
|x-7| > 3

Solve: |x+7|+15>12

|x+7|+15 > 12
|x+7|+15 > 12-15
|x+7| > -3
which is true for all real numbers because an absolute value expression is always greater than any negative number.

how do I write in inequality form : all real numbers at most 6 units from 0 ?

|x-6| < 0

Solve: 7-4|x+7|>-5

7-4|x+7|>-5
-4|x+7| > -5-7
|x+7| < -12/(-4)
-3 < x+7 < 3
-10 < x < -4

A number n is no more than 7 units way from 5. Write an absolute-value inequality to show the range of possible values for n.

so that

For what value of x is this inequality |x – 6|< - 8 true? Why?

for no value of x... 
|x – 6|< - 8 
has no solution because an absolute value of a real valued expression can never (by definition) be negative. Therefore it can never be less than -8

Solve the inequality. Write the solution set in interval notation. |5 - x| ≥ 3

 
means 
 
or 
 
which lead us to 
 
or
 

ie 
(-oo,2)U(8,oo) 

Solve the absolute value inequality

 

 

which means 

 


 

A Steinway piano should be placed in a room where the relative humidity h is between 38% and 72%. Express this range with an inequality containing an absolute value.

We are essentially converting the inequality 

 

into an absolute value inequality 

The middle of the interval is 

 

The halfway distance on the interval is 

 


therefore our interval is 

 


:)

2+|4x-6|<12 

|4x+6|<10 

-10 < 4x+6 < 10 

-10+6 < 4x < 10+6 

-4/4 < x < 16/4 

-1 < x < 4

Find the interval notation for |2x-3|>-3 and its graph.

The statement 
|2x-3|>-3 
needs no work at all. An absolute value by definition is always greater than or equal to 0 (and therefore greater than any negative number) 
The graph is the entire number line and the interval is (-oo,oo). 


:)

Solve 7+5|c|≤1-3|c|

 


 

 

W can stop immediately because the absolute value can never be negative so this inequality has no solution 

:)