[SOUND] Let's learn how to solve a linear inequality.

[SOUND] For example, let's solve the following inequality for v and write our

answer in interval notation. Now, for the most part, when we solve

linear inequalities, we proceed in the same way as we do when we solve linear

equalities. However, there is one difference, which

we'll see. Let's begin by getting the negative seven

to the right-hand side or we'll add seven to each side.

In other words, we have -7 + 7 + 4v is greater than or equal to -15 + 7.

And -7 + 7 = 0, they'll cancel and we're left with 4v greater than or equal to

negative eight. Now, let's divide both sides by four,

which is exactly the point we need to be careful.

This is where the difference between inequalities and equalities comes up.

If this number four would have been a negative number, we would have had to

flip or reverse the direction of that inequality.

[SOUND] In other words, in general we have the following, that when we multiply

or divide both sides of an inequalilty by a positive number, the inequality stays

the same, like in this case. However, when we multiply or divide by a

negative number, we flip the inequality sign.

Okay. So we don't have to flip our inequality here, so we have v greater

than or equal to negative two. Let's look at this on a number line.

Here is negative two, v can equal negative two and anything to the right.

Now, we are asked to put our answer in interval notation.

Doing this, we have closed bracket negative two, because we want to include

negative two up to positive infinity, which would be our answer.

Alright. Let's see another example. [SOUND] Let's solve the following

inequality for x and write our answer in interval notation.

Again, we'll begin by getting this number -15 to the right or adding 15 to both

sides. So we have - 3x - 15 + 15 > 3 + 15.

And -15 + 15 is 0, they'll cancel, and we're left with - 3x > 18.

And now, let's divide both sides by negative three, however, remember,

[SOUND] that when we divide both sides of an inequality by a negative number, we

have to flip this inequality sign. That is, we need to flip the greater than

and make it a less than, so we get x < -6.

Okay. Let's look at this again on the number

line. It's negative six. We do not want x to

equal negative six, so we put an open circle and then less than everything to

the left of negative six. And again, we're asked to put our answer

in interval notation, therefore, our answer is negative

infinity up to negative six open parentheses because we do not want x to

equal negative six. Now, if we didn't want to worry about

dividing by this negative, we could have started just a bit different here.

We still have the same inequality, - 3x - 15 > 3,

but then, if we bring the - 3x to the right or add 3x to both sides, and bring

the 3 to the left or subtract 3 from both sides, what do we get? We have - 15 - 3 >

+3x or -18 > 3x. And, now when we divide this number's

positive so we don't have to worry about the inequality flipping.

So we get x < -6, which we can, right in the other direction,

x < -6, which is our, same answer.