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|k-4|>1 I don't get how to solve this problem graphically

I'm confused onchow to solve inequalities graphically 

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4 Answers

Hi Karen,

Maybe this will help. Let's break it down into little pieces so bear with me! First, go by the DEFINITION of the absolute value. 

|x| = x if its positive or -x if x is negative. So, lets consider something simple. |x| = 3. You know that x = 3  and -3. So, you know that when dealing with absolute values, there will be 2 solutions instead of 1.

Now I'm not limited to using the absolute value on a single variable x. I can also take the absolute value of somehting more complicated like k - 4. Replace x with k - 4.

|k - 4| = k - 4 and -(k-4). So, I have two solutions just like my simple example above. The rest of the problem states that |k-4| > 1 right? Well, if since |k-4| = k-4 then k-4 > 1 AND |k-4| = -(k-4) then -(k-4) > 1. We have two inequalities because the absolute value has two solutions.

Now, let's solve for k in both equations. You get k > 5 and k < 3. k is just a label for the real number line. The two inequalities say that |k - 4| > 1 will be true if k > 5 and k < 3. You draw the number line and cover the numbers greater than 5 and the numbers < 3 with a thick line. Since the inequality is just greater than, you have to show that k = 5 and 3 are NOT part of the solution. You can do this by starting the thick lines at open circles that lie at k = 3 and 5.

It was long but I want you to understand how the absolute value works. Hope it helps!



An excellent explanation!

Your definition of absolute value is simple and exact.  [k-4] = (k-4) AND -(k-4)>1 is the key. From that point on everything you say is right on the money! I especially like your explanation of use of open circles showing that k = 5 AND 3 are not included as part of the solution.  GREAT WORK!

l X - 4 l > 1
  X - 4 > 1    or   X - 4 < 1
   X > 4     X < 5
            <_______________-5 -4 -3-- -2 -1 ---0---+1--+2---+3---+4_______________>. 
 The full line on graph line is solution. Notice how to handle inequality with absolute value.

Um, Helen, this is a single variable, a number line would be right, wouldn't it?

So, draw your numberline.

Now consider the geometric properties of the absolute value.  Your statement says (and watch this carefully!) "The distance between x and 4 on the number line is greater than 1"

So find 4 on your numberline.  Go one number to the right, draw your circle, and put an arrow indicating distances that are all greater than one.

Now go one number to the left, draw your circle, and put another arrow indicating all DISTANCES that are greater than one.  DISTANCES, NOT NUMBERS.  Hint: This arrow will be in the opposite direction of the first one.

Usually, this type of question is asking you to draw an x-y axis on paper (graph paper is best) and use shading to indicate where on the graph (a region of values above or below a line) make the inequality true. Dashed lines indicate "less than" or "greater than" while solid lines indicate "less than or equal to" and "greater than or equal to".