00:00Absolute value within absolute value x, minus x, plus x, less than or equal to 4.
00:05The question asks us to find the sum of the integer values of x that satisfy the inequality.
00:10Now, when I look at it, the absolute value within the absolute value of x, minus x, plus x, is less than or equal to 4.
00:18Now, when I look at the innermost absolute value, I mean, right here, folks, what's the critical point of x?
00:25x equals 0, right? We'll think in terms of x equals 0, folks.
00:29First, let's say x is greater than or equal to 0, so that's what we'll examine.
00:36So what will x friends be when they leave here?
00:38It comes out as plus x, right? From the absolute value of x, x comes out.
00:42Yes, x minus x plus x is less than or equal to 4.
00:49The x's cancel each other out, so we directly find x to be less than or equal to 4.
00:54But how is it less than or equal to 4? What was our first condition?
00:57x was greater than or equal to 0, right guys?
00:59Then the range of our x in the first condition must be less than or equal to 4 and greater than or equal to 0.
01:06Looking at the second one, friends, we'll examine it for x less than 0.
01:14If x is less than 0, how do we find the absolute value of x?
01:18Since x is less than zero, we subtract it by putting a minus sign in front of it, minus x.
01:22Minus x, minus x, plus x.
01:26It's less than or equal to 4, right guys?
01:28So, what happened now?
01:30Minus 2x, plus x, less than or equal to 4.
01:35So, if -2x means x is less than zero, then -x is greater than zero.
01:42When we multiply everything by a negative number, right guys?
01:44Inequality is within us, and our inequality changes direction.
01:49So, when we multiply by 2, minus 2x becomes greater than 0.
01:54If -2x is greater than zero, then our expression here comes out exactly as it is.
02:00It comes out as minus 2x.
02:01Because it's greater than zero.
02:02It doesn't get a negative mark in front of it.
02:04It comes out exactly as pictured.
02:05Plus x is less than or equal to 4.
02:08What happened, guys?
02:11What happened to minus x, guys?
02:13It became less than or equal to 4.
02:16What happened to our X, guys?
02:19Greater than or equal to minus 4.
02:21So, what was our x here, guys?
02:24It was less than zero, wasn't it?
02:26Yes.
02:27For those of you who are younger than zero, this is how we write it.
02:30What is your solution set in this case?
02:35The gaps are zero to minus 4, open gaps, and zero to 4 closed gaps, my friends.
02:46So, in short, what do -4 and 4 mean, guys?
02:52Our solution lies entirely within the closed range.
02:56So, what are the integer values of x in this case?
03:00Let's take a look.
03:00Minus 4, it asks for the sum of these.
03:03I'm going straight to addition.
03:04Minus 4, plus, minus 3, plus, minus 2, plus, minus 1, plus, zero, plus, 1, plus, 2, plus, 3, plus,
03:13I wrote a long explanation of number 4.
03:14When we add it all up, friends, we get the result is zero.
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