00:01
If 2 is less than x is less than 6, which of the statements about x are necessarily true, and which aren't? okay, for this problem, let's draw a number line.
00:18
So here we have two.
00:21
This is going to be an open circle because it's less than.
00:24
It's not less than or equal to you, so it doesn't include two, and an open circle on this side.
00:30
And this is everywhere that x can be.
00:35
X is less than 2 is less than 6.
00:39
Okay, so now which other statements are true and which aren't necessarily true? okay, let's start with a.
00:49
Zero is less than x is less than 4.
00:55
So based on this concept of x being stuck between 2 and 6, can this be true? can x be one here? x can be one in part a, but it can't be true here in our given expression.
01:12
So this is not necessarily true.
01:21
Okay, how about b? x0 is less than x minus two is less than four.
01:32
Yeah.
01:33
Okay, so what does this mean? let's solve to isolate this x what that means is just adding two to each part of this inequality to get x by itself and we get two is less than x is less than six this is true this is exactly what our expression says it's exactly what it says two is less than x is less than six so this is necessarily true okay part c one is less than x over two is less than three what do we do with this one this one this one we're going to isolate that x again.
02:16
So here we do that by multiplying every part of the inequality by two to get rid of that two here.
02:24
So once we do that, two times one is two, two times x over two is just x, two times three is six.
02:32
This is true.
02:34
This relates back to our given expression of two is less than x is less than six.
02:39
So that is true.
02:40
Okay, now part 1 6th is less than 1 over x is less than 1 half.
02:53
Okay, this one's kind of annoying, isn't it? what can we do here? well, one thing that we can do is try to clear these denominators by multiplying everything by 12x.
03:14
Let's just see how that works out.
03:17
12x times 12x over 6 is 2 12x over x is 12 12 12 x 2x sorry 12 x over 6 is 2x 1 x 12 12 12 x over 2 is 6x okay what does that mean let's divide by these 2s to see if we can get this any clearer this means x x is greater than 3 is greater than 2x.
03:52
What the heck does this mean? what does it mean? so it means that x is less than 3 and x is less than 2x if we think about it that way.
04:04
X is less than 3, which is less than 2x.
04:09
So let's look back at this and let's just plug in a number for x to see if it's true, something that fits here.
04:15
So like the number 3 is 3 less than 3 less than 6.
04:21
Well, no, not this isn't true.
04:25
And the main reason that this isn't true is because 3 is not less than 3.
04:31
So if there was a less than or equal to sign, this would work fine, but there isn't one.
04:36
So this is not true.
04:39
Well, what about if i plug in 2? 2 is less than 3 is less than 4, that works.
04:43
But for any number bigger than 3, it doesn't work.
04:48
And also if we just look at this, the only, x has to be a number, be between 2 and 6, right? and 3 would be right here.
04:58
So there's going to be very, very little overlap between this and this.
05:04
So that is not true.
05:07
Okay, part e, 1 is less than 6 over x is less than 3.
05:17
For this, let's try that method of plugging in the lowest, so x equals 2.
05:22
Remember, it can't actually be equal to two.
05:25
All i'm doing is testing the waters here to see if it works.
05:28
We'll get one is less than 6 over 2 is 3...