Tip:
Highlight text to annotate it
X
>> All right.
We're going to use the properties of the inequalities to solve some linear inequalities.
So let's look at this first one.
3x-4 is less than or equal to -10.
All right, now remember to solve an inequality, it's exactly the same rules
as solving equalities except for one caveat, when you multiply or divide
by a negative number, both sides of the equation by a negative number,
then you have to switch the inequality,
only when you multiply or divide by a negative number.
All right, so how would we solve this?
We solve it similar to an equation.
And I see 3x-4.
If that was an equal sign, we would add 4 to both sides,
and I'm just gonna write over here what I'm doing.
I'm adding 4, so I'm not even multiplying or dividing by anything,
so we can keep the inequality as is to get 3x is less than or equal to -6.
All right, next thing I want to solve for x, I'm gonna have to divide both sides by 3.
I divided both sides by 3.
Okay, that's okay.
You could always multiply or divide by a negative number and preserve the solution set,
so this is x is less than or equal to -2.
All right.
So remember, this means there are a lot of solutions, right?
So let's see, if there's 0, and let's say, there's 2.
And here's -2.
All right, it says x is less than or equal to -2, so that means that 2 is a solution
and numbers to the left of 2 are also solutions.
So, I could put a dot on -2 because that's part of the solution, as well as go to the left.
So this represents infinitely many numbers as solutions.
You can't list them all, right?
Like, there's -3.
-4. -2 1/2, etcetera.
Okay, now what we want to do is check this and make sure we've got it going
in the right direction, and to make sure really the dot should be at -2.
So it's really a two-step process to check it.
So first of all, let's check that 2 makes -- is a solution, right, because I've got a dot on 2.
So first let's check to make sure x is 2 is a solution, into the original inequality.
All right, so what we're going to do is plug in 2 for x, and we have six --
I'm sorry, I'm plugging in -2, so that's gonna be -6-4,
and -6-4 is -10, and the other side's -10.
And what I want, I want to make sure that 10 is less than or equal to -10.
Actually, it's equal, right?
It says it could either -- either both sides could equal, or the left side has to be less
than the right-hand side, so 2 is certainly a solution.
Now, we also have to check that a number to the left of 2 is a solution,
or we could check that a number to the right of 2 is not a solution.
You can do either way.
So let's continue, all right?
Let's say, the original problem, again, was 3x-4 is less than or equal to -- I think it was -10.
All right, so let's check that some number to the right is not a solution.
I usually like to check 0 because it's super easy to plug in for x.
So x equals 0 should not be a solution, right?
Keep that in mind.
It should not be a solution, so let's check 0.
I'm just gonna plug in 0.
So we have 3 times 0-4.
-4 on this side, and the right side is -10.
And -4 is less than -10?
No, -4 is bigger.
Perfect. We didn't want it to be a solution,
so it's good that the solution set is not going through zero.
So therefore, this is the correct solution.
So this is just a check.
Now, if you wanted to check something else, you can check to see that x equals, let's say, -5,
for instance, that's a number that should be a solution, so let's see what if I plug in -5?
That should make it a solution.
So, if we plug in -5 for x, we have -15-4, which is -19 on this side.
And is -19 less than -10?
Yes. Perfect.
So again, what we had originally is correct right here.
x is less than or equal to -2, and if I were asked
to graph it, this is what it would look like.
Okay, let's look at another example.
We have -2/3x is less than or equal to 2.
All right, we could solve this in more than one way, but the usual way when we have fractions --
one way is to just multiply both sides by the reciprocal.
So the reciprocal of -2/3 is -3/2.
Right. So if I want, I can say, "Well, I'm gonna multiply both sides by -3/2.
So on both sides I want to multiply by -3/2.
So I'm gonna write down what I'm doing.
I'm multiplying by -3/2 -- oh, wow.
I multiply the inequality by a negative number; make sure you switch the direction here.
Now, here's how I do it.
If I do it right at this step, I cross it out right away, so I don't forget.
I'm gonna write -- oh, now, it's really got to be greater than or equal.
Okay? So look at how it was originally.
It was just, in what's in black, the -2/3x is less than or equal to 2, and if I don't want
to take another step to write this -3/2 on both sides, then make sure that you cross
that off and switch it right away.
So what we have, then, on the left-hand side, everything cancels, so I have x, and that's,
of course, what I'm trying to solve for, but it's greater than or equal to.
Now, let's see what happens over on this side.
I've got positive times a negative, so the answer's negative.
The 2s cancel, and I got an answer of -3.
So it looks like this has a solution set of numbers greater than or equal to -3.
Again, since it's equal, we can put a dot on that to show that's part of the solution.
And it's going to the right.
Now, it's going through 0, so that means 0 should be a solution,
so I'm gonna check the original, the original problem.
I'm gonna check to make sure x equals 0 is a solution in the original,
-2/3x is less than or equal to 2.
And if I put in 0 for x, ha, nice, -2/3 times 0 is just 0, and on the left side I have 2,
and you want to check to make sure it's less than.
X equals 0 should be a solution because it's going through that number.
And you can also check -3.
When you plug in -3, you should get the same number on both sides.
That's how you make sure that you've got the starting point in the right place.
So if you plug in -3, notice that you would have 2 equal to 2.
I'm gonna leave that to you.
So the solution is x is greater than or equal to -3.
The solution set is -- I mean, I'm sorry, the graph is shown here.
All right, now I'm gonna show you what would happen if you, instead of doing it
on this same step, you had gone ahead
and took a second step before you multiplied by negative 3/2.
Okay, instead of right at that point, doing it times -3/2, you could just always come
down to the next spot and multiply both sides by -3/2.
It's a little bit less messy, and then at the same time,
you're gonna write the greater than or equal at that point.
And then you would have this; times both sides by 2,
and you would still get the same answer, x greater than, equal to 3.
Last way to do it.
-2/3x less than or equal to 2.
You could just take the opposite of both sides, so that it's a positive 2/3,
but that means you also have to change the directions at the same time,
just like we did with inequalities -- I mean, with equations, and then solve it.
There are some other approaches you could use to solve this inequality.
One of them is to simply multiply both sides by the least common denominator,
since there's a fraction involved.
So, if you would have done that, the least common denominator would be 3,
and so you could just multiply both sides of the inequality by 3, but remember,
this is a positive number, so you don't need to switch the inequality symbol.
And so this would give you -2x is less than or equal to 6.
Now you could divide by -2, and if you do it at this step, though, remember,
if you put divide by -2 right here, this is no longer valid.
It's no longer less than or equal.
You would have to change that to a greater than or equal right there.
Don't wait until the next step to do it because this would be a false statement at this point.
So this gives you x is greater than or equal to -3, like we got before.