00:00absolute value of 2 times x minus 2 plus absolute value of 1 minus x divided by 1 plus absolute value
00:05inside 1 minus x.
00:07The question asks us to find how many integer values of x do not satisfy the greater than 1 inequality.
00:13Before we move on to that, I want to say something.
00:17Let's write 2 times x minus 2 as one.
00:21Plus 1 minus x divided by 1 plus 1 minus x.
00:28It's greater than 1.
00:29I want to pass this 1 to the other side, guys.
00:32This is more accurate.
00:33Actually, this means everything will be absolute values, meaning it will be positive, but let's show it anyway.
00:38absolute value of 2 times x minus 2 plus 1 minus x divided by 1 plus absolute value of 1 minus
00:47x.
00:48Guys, I know now that we can pull out those two, right?
00:53We will become familiar with these now.
00:54Here's how we'll do it, guys.
00:55Can I write it as 2 times x minus 1?
00:59Yes.
01:00Can I write 1 minus x in negative form?
01:02So, can I write it as x minus 1?
01:04When I multiply it, it still gives me minus 1 minus x.
01:07It's essentially the same expression because absolute value swallows it up.
01:11I'm saying plus x minus 1.
01:14Divided by 1 plus x minus 1.
01:16I wrote them all in the same style.
01:17I also added -1 here.
01:20Greater than 0.
01:20Now I can move on to my point about inequality, folks.
01:23What is 2 plus one, guys?
01:253.
01:253 times x minus 1 divided by 1 plus x minus 1.
01:32Minus 1 is greater than 0.
01:34If we multiply it like this and add this place as a friend...
01:373 times x minus 1 minus 1 minus x minus 1 divided by 1 plus x minus 1 is greater than 0.
01:50Now, friends, if you subtract one x - 1 from three x - 1s, what's left?
01:542 times x minus 1.
01:57Minus 1 divided by 1 plus x minus 1.
02:01It's bigger than 0.
02:02Okay guys, I know that for the numerator to be greater than 0, either the numerator or the denominator must be negative.
02:10the denominator cannot be negative.
02:12In this situation, only one option remains.
02:14Both the numerator and the denominator are positive.
02:18In other words, the state of being positive.
02:20And I already know that, it's a capital 0.
02:22Why a capital 0, friends?
02:24If we add 1 plus something with an absolute value, what do we get?
02:26It comes out as an expression greater than 0, right?
02:29Then the top part also needs to be a capital 0.
02:31So, 2 times x minus 1 minus 1 must be greater than 0.
02:40So what will happen to the friends then?
02:422 times x minus 1 is greater than 1.
02:45We'll move -1 to this side.
02:471.
02:47I divided each side in two.
02:49The expression we call the absolute value of x minus 1 is actually greater than 1 divided by 2.
02:56When we do this, then friends, two options will appear from here.
03:01Let me do this.
03:021 or x minus 1 is greater than 1/2, which means x is greater than 3/2, right guys?
03:10Yes.
03:10Or 2x minus 1 is less than minus 1 divided by 2, and if we move that minus 1 to this side...
03:17What is our x, guys?
03:19It is less than 1/2.
03:21So our solution is less than -1/2, the open-spaced union from 1/2 is 3/2 as you can see.
03:41infinitely wide open.
03:44Or you could write it like this, which is still relevant to our current question.
03:49All real numbers have a closed interval between the difference (the part to be subtracted) of 1/2 and 3/2.
03:57So, the part we extracted contains the integer values of x that don't satisfy the condition.
04:04So, what integers do we have between 1/2 and 3/2?
04:11Isn't there only one, guys?
04:13So, could we actually say it like this?
04:16Our solution set is equal.
04:20Equals the difference of all real numbers.
04:24What do we subtract from the integer values that don't currently match integer values?
04:29We'll remove 1.
04:31So, in fact, we have one integer value that doesn't satisfy the condition.
04:35When you put that in its place, you get 0.
04:38The only integer value for x is 1.
04:40There is only one.
04:44Our only missing value is 1.
04:46We have values that we don't uphold.
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