Lesson BASICS - Absolute value
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<b>Introduction</b> 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: {{{graph(200,200,-6,4,-4,6,x+2) }}} whereas eg y=|x+2| is: {{{graph(200,200,-6,4,-4,6,abs(x+2)) }}} <b>Algebraic meaning</b> 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: {{{graph(200,200,-6,4,-4,6,abs(x+2), 3) }}} Algebraically we have x+2 = 3 --> x = 1 and -(x+2) = 3 -x-2 = 3 -x = 5 --> x = -5 ---------------------------------------------------------------------- <b>An Interesting Example</b> 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 {{{ graph(300,300,-4,4,-6,6,x-4,2x-2) }}}. The solution is where they cross. Now, plotting the absolute versions, we see {{{ graph(300,300,-4,4,-6,6,abs(x-4),abs(2x-2)) }}}. 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) ---------------------------------------------------------------------- <b>Further Examples</b> *[problems 28917/15843] *[problems 29670/16402]
