1
00:00:02,867 --> 00:00:06,333
We're asked to solve for x and we have this compound inequality here. Negative sixteen is less than or equal

2
00:00:06,333 --> 00:00:11,933
to three x plus five which is less than or equal to twenty. And really there's two ways to approach it which are

3
00:00:11,933 --> 00:00:16,400
really the same way and I'll do both of them. And I'll actually do both of them simultaneously. So

4
00:00:16,400 --> 00:00:20,933
one is to just solve this compound inequality all at once. I'll just rewrite it. Negative sixteen

5
00:00:20,933 --> 00:00:26,600
is less than or equal to three x plus five which is less than or equal to twenty. And the other way

6
00:00:26,600 --> 00:00:32,733
is to think of it as two separate inequalities but both of them need to be true. So you could also view

7
00:00:32,733 --> 00:00:43,533
it as negative sixteen has to be less than or equal to three x plus five, and three x plus five needs

8
00:00:43,533 --> 00:00:49,933
to be less than or equal to twenty. This statement and this statement are equivalent. This one may

9
00:00:49,933 --> 00:00:54,733
seem a little bit more familiar because we can independently solve each of these inequalities and just

10
00:00:54,733 --> 00:00:59,133
remember the AND. This one might seem a little less traditional because now we have three sides to

11
00:00:59,133 --> 00:01:02,800
the statement. We have three parts of this compound inequality. But we can see that we are actually going

12
00:01:03,467 --> 00:01:09,867
to solve it the exact same way. In any situation we really just want to isolate the x on one side of the inequality,

13
00:01:09,867 --> 00:01:14,867
or in this case one part of the compound inequality. Well the best way to isolate this x right here

14
00:01:14,867 --> 00:01:19,267
is to first get rid of this positive five that is sitting in the middle. So lets subtract five from

15
00:01:19,267 --> 00:01:27,267
every part of this compound inequality. Subtract five there, subtract five there, and subtract five over there.

16
00:01:27,267 --> 00:01:34,467
So we get, negative sixteen minus five is negative twenty-one, is less than or equal to 3x plus

17
00:01:34,467 --> 00:01:41,467
five minus five is 3x, which is less than or equal to twenty minus five which is fifteen. And we can essentially

18
00:01:41,467 --> 00:01:46,800
do the same thing here. If we want to isolate the 3x, we can subtract five from both sides, subtract

19
00:01:46,800 --> 00:01:53,933
five from both sides, we get negative twenty-one. Negative twenty-one is less than or equal to 3x.

20
00:01:53,933 --> 00:02:03,200
And we get, subtracting five from both sides, and notice we are just subtracting five from every part of this compound inequality.

21
00:02:03,200 --> 00:02:09,600
We get 3x is less than or equal to fifteen. So this statement and this statement are once again the

22
00:02:09,600 --> 00:02:16,533
exact same thing. Now going back here if we want to isolate the x, we divide by three. We have to do

23
00:02:16,533 --> 00:02:20,800
it to every part of the inequality. And since three is positive we don't have to change the sign.

24
00:02:20,800 --> 00:02:27,533
So let's divide every part of this compound inequality by three. This is equivalent to dividing every

25
00:02:27,533 --> 00:02:35,867
part of each of these inequalities by three. And then we get, negative twenty-one divided by three is negative

26
00:02:35,867 --> 00:02:43,933
seven, is less than or equal to x, which is less than or equal to fifteen divided by three is five.

27
00:02:43,933 --> 00:02:53,467
You do it over here, you get negative seven is less than or equal to x, and x is less than or equal to

28
00:02:53,467 --> 00:02:59,067
fifteen over three which is five. This statement and this statement are completely equal. Now we've

29
00:02:59,067 --> 00:03:05,467
solved for x. We've given you the solution set. And if we want to graph it on a number line it would

30
00:03:05,467 --> 00:03:14,933
look like this. This is zero, this is five, this is negative seven. Our solution set includes everything

31
00:03:14,933 --> 00:03:21,933
between negative seven and five, including negative seven and five. So we have to fill in the circles

32
00:03:21,933 --> 00:03:28,933
on negative seven and positive five, and it is everything in between. That's our solution set.

33
00:03:28,933 --> 00:03:32,133
And so we can verify that these work. You can try out a number that's well inside

34
00:03:32,133 --> 00:03:38,400
of our solution set like zero. Three times zero is zero. So you're just left with five is greater

35
00:03:38,400 --> 00:03:42,400
than or equal to negative sixteen, which is true. And five is less than or equal to twenty.

36
00:03:42,400 --> 00:03:45,667
Or negative sixteen is less than or equal to five, which is less than or equal to twenty.

37
00:03:45,667 --> 00:03:51,267
So that works, and that makes sense. You could try five. If you put five here you get three times five

38
00:03:51,267 --> 00:03:55,533
plus five, well that's just twenty. Negative sixteen is less than or equal to twenty, which is less

39
00:03:55,533 --> 00:03:58,800
than or equal to twenty. That works. Negative seven should also work.

40
00:03:58,800 --> 00:04:03,400
Three times negative seven is negative twenty-one plus five is negative sixteen. So you get negative

41
00:04:03,400 --> 00:04:08,800
sixteen, which is less than or equal to negative sixteen, which is less than or equal to twenty.

42
00:04:08,800 --> 00:04:13,133
And you could try other values. You could go outside of our solution set. Try something like ten.

43
00:04:13,133 --> 00:04:17,800
Ten should not work. If you put ten here, you get three times ten plus five is thirty-five.

44
00:04:17,800 --> 00:04:23,667
Negative sixteen is less than or equal to thirty-five, but thirty-five is not less than or equal to twenty.

45
00:04:23,667 --> 99:59:59,999
And that's why ten is not part of our solution set.