Lesson BASICS - Absolute value

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
please help
See answer to question 28917

Answer #15843 by longjonsilver(2297)   You can put this solution on YOUR website! the graph of y=|x+4| looks like: it is just the graph of y = x+4 except where we have negative y-values...here the line is reflected back...the line is essentially y=-(x+4) So, look at the V-shaped line. We need to know where this is less than 6... draw a horizontal line at y=6 and you get the answers: OK, algebraically, we have in essence 2 equations to compare with y=6. 1. x+4<6 --> x < 2 is one answer (which matches with the graph) 2. -(x+4) < 6 --> -x - 4 < 6 --> -x < 10 --> x > -10 is the other answer (which matches with the graph too). jon.

Question 29670: |10x+60|=|x+90|
See answer to question 29670

Answer #16402 by longjonsilver(2297)   You can put this solution on YOUR website! The graphs look like . Homing in on the 2 solutions, we see that which show the solutions to be about x=4 and x=-14. Roughly. So, algebraically: 10x+60 = x+90 9x = 30 x = 30/9 x = 3.333333 10x+60 = -(x+90) 10x+60 = -x-90 11x = -150 x = -150/11 x = -13.636363.. jon