Absolute Values

The absolute value of a number is the value of the number ignoring whether the number is positive or negative.
Eg.
|9| = 9
|-3| = 3
|-x| = x

Absolute value expressions can be shown as a graph.
A y = |x| graph is basically a y=x graph where all the negative points are placed on the positive side.
y=x

y=|x|

This y=|x| can be described as being a y=x graph when x>0 and a y=-x graph when x<0

You may remember that a parabolic expression in the form of

Will be a y=x^2 with the axis on (k,h) and will be thinner or wider depending on the value of a.
This is the same case with absolute value graphs.
Eg, y = 2|x-2|+1

y = -|x+1|+2

Composite Absolute value graphs
Graphs such as y=|x+1|+|x-2| or y = |x-2|-|x+3| will require more effort to graph.
I will explain how to sketch these graphs as well as provide some examples below.

Addition of absolute value graphs.
Eg1. y=|x+1|+|x-2|

If you were to sketch both the graphs of y=|x+1| and y =|x-2| you may notice that

1. When x< -1   |x+1| is in the form of -(x+1) and |x-2| is in the form of -(x-2)
2. When -1<x<2    |x+1| is in the form of x+1 and |x-2| is in the form of -(x-2)
3. When x>2    |x+1| is in the form of x+1 and |x-2| is in the form of x-2

If you were to add up the 2 graphs from each of the 3 areas you would find

1. When x<-1  -(x+1) +  -(x-2) = -2x + 1
2. When -1<x<2  x+1 -(x-2) = 3
3. When x>2  x+1 + x-2 = 2x-1

Now that you know what the graph would look like in each section you just graph the 3 sections

Eg2. y = x+|x-2|

After sketching the two graphs you would notice

1. When x<2 y=x is in the form of y=x, y=|x-2| is in the form of -(x-2)
2. When x>2 y=x is in the form of y=x, y=|x-2| is in the form of (x-2)

After adding the resolving each section

1. When x<2  y = x - (x-2) = 2
2. When x>2  y= x + x-2 = 2x-2

After sketching each of the two parts it should look like this:

Subtraction of absolute value graphs

The concept for subtraction is the same except instead of adding we are subtracting.

Eg1. y = |x-2|-|x+3|

1. When x<-3  |x-2| is in the form of -(x-2)  |x+3| is in the form of -(x+3)
2. When -3<x<2  |x-2| is in the form of -(x-2)   |x+3| is in the form of (x+3)
3. When x>2  |x-2| is in the form of (x-2)   |x+3| is in the form of (x+3)

Therefore 

1. When x<-3   -(x-2) - -(x+3)= 5
2. When -3<x<2   -(x-2) - (x+3)= -2x -1
3. When x>2   (x-2) - (x+3) = -5

The graph would look like this

Absolute Value Equations and Inequalities

Eg1. |2x+7| =3

To solve an equation like this we have to break up the absolute value. |a| = |-a|
Therefore we can say that if |2x+7| = 3  2x+7 = ±3
Then we can just solve for 2x+7 =3 and 2x+7 = -3
So x = -2, -5
Before we say that the answer is x=-2 or -5 we must substitute those two values back into the original equation before there are special cases where the solutions may be invalid.
In this question both solutions are valid.

Eg2. |2x+3| = |x+6|
2x + 3 = x+6  or 2x+3 = -x-6
x=-3 or 3
Both solutions are valid.

Eg3. |x+2| =2x+1
x+2 = 2x+1  or x+2 = -2x-1
x = 1 or -1

When x = 1 LHS = 3 RHS = 3
When x = -1 LHS = 1 RHS = -1
Therefore x=1
This question shows that there are cases where the solutions may be invalid

Eg4. |2x+5|<7
Absolute value inequalities can be solved using test point.
Consider |2x+5|=7
2x+5 = 7 or 2x+5 = -7
x = 1 or -6

Those 2 points will be the points when LHS = RHS in this case we want LHS<RHS so we test a point for each of the 3 regions:

1. x<-6  False
2. -6<x<1  True
3. x>1  False

Therefore the inequality is true for when -6<x<1

This concludes my post.

Absolute values is actually quite a simple topic if you know the fundamentals.

Hope you have learnt something