# Lesson BASICS - Absolute value

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 Algebra: Absolute value Solvers Lessons Answers archive Quiz In Depth

This Lesson (BASICS - Absolute value) was created by by longjonsilver(2297)  : View Source, Show
About longjonsilver: I have a new job in September, teaching

Introduction
This Lesson details the algebra surrounding the "absolute value" equation, sometimes called "modulus".

eg solve |x+2| = 3

Before we do this, I shall explain what the absolute value actually means.

|x+2| means the straight line y=x+2 is broken at the point where the line becomes negative. The negative portion of the line is reflected back up, thus creating the classic V-shaped graph.

eg y=x+2 is:

whereas eg y=|x+2| is:

Algebraic meaning
The graph is actually 2 straight lines, one is y=x+2 and the other is y=-(x+2).

So this is all you have to remember...there are 2 equations and to find the second one, we place a minus sign around the entirity of the original.

so, let us answer the above example: solve |x+2| = 3

From what i have told you, and your understanding of what the |x+2| graph looks like, it is asking where does it equal 3? Well, drawing a horizontal line ay y=3, will give you 2 points. We can see them on the following graphical answer:

Algebraically we have
x+2 = 3 --> x = 1

and
-(x+2) = 3
-x-2 = 3
-x = 5
--> x = -5
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An Interesting Example

Solve |x-4| = |2x-2|.

OK, lets look at some graphs first, to highlight the answers we expect.

Plotting y=x-4 and y=2x-2 gives . The solution is where they cross.

Now, plotting the absolute versions, we see .

Notice how there are 2 solutions now. One is the "mirrored" version of the first graph and the other is a new solution.

So, algebraically, we have:

1st solution is when x-4=2x-2
--> -4 = x-2
--> -2 = x
ie when x=-2

when x=-2, we have y=|x-4|
y = |-2-4|
y = |-6|
y = 6

--> solution 1 is (-2,6)

2nd solution is when x-4=-(2x-2)
--> x-4 = -2x+2
--> 3x-4 = 2
--> 3x = 6
ie x = 2

when x=2, we have y=|x-4|
y = |2-4|
y = |-2|
y = 2

--> solution 2 is (2,2)

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Further Examples

Question 28917: I am totally lost,
I'm suppose to solve the inequality and then graph
absolute value x+4 end of absolute value less than 6