
Title:
Solve for x  College Algebra

Description:

So here, I have redrawn our number line, and for each region, I've picked out a

value to test to see whether it satisfies our inequality. First, let's test x

equals negative 11. Now, you might think that we need to actually plug negative

11 in and find out all the numbers here, but all we really care about is the

sign of this entire expression. We just want to know whether or not it's

positive. So, what I'm going to do is just think about whether or not each of

the factors here, we have 3 of them, is positive or negative. And then, of

course, I also need to take into account the negative sign that's on the

outside. So, I'll write that one first. We have a negative sign on the outside.

If I plug in negative 11 here, negative 11 plus 10 is a negative number, so we

have negative and negative in the numerator. And then in the denominator, we

have a negative number for this factor and a negative number for this factor.

Four negatives multiplied or divided together, gives us a positive number. So,

this region works. Now, let's do the same thing for negative 5. And going

through the same sorts of steps for the next 3, it turns out that only this

region, the first one you found in the area between negative 2 and 2, yield

positive solutions. Let's check the 3 critical values this way also. Notice that

when we plug in negative 10, we end up with 0 for the numerator. That means,

that this entire fraction can never be greater than 0. So, negative 10 is not

part of our solution set. Negative 2 and positive 2 each make the expression on

the lefthand side contain a division by 0, so these cannot be part of our

solution set either. Great. So, we have these two intervals to take into

account. We have the interval negative infinity to negative 10, united with the

interval negative 2 to 2. Awesome. That's our solution. Great job with some

really tough work on quadratic inequalities and rational inequalities. I know

there are a lot of steps in all these problems, but I helped you develop a

little bit of independence in figuring out how to solve these kinds of

inequalities. They're pretty cool, too. I'm going to do one more quick thing and

show you graphically what this inequality might look like.