I'm confused onchow to solve inequalities graphically
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!