Solving Rational Inequalities

[lolily]

In my textbook, there's an example inequality problem that looks like this: -1/(x+5) ≥ -3x/(x+5), and the instructions simply say to solve. I understand that you have to get zero to one side, which gets you 0 ≥ (-3x+1)/(x-5), and then set the numerator and the denominator equal to zero to find the endpoints, x=1/3 and x=5.

Now, the next part is what I don't understand. The example problem says that we get the intervals (-∞,1/3], [1/3,5), and (5,∞), but then says that we must check the intervals one by one to see which factors agree in sign. Here's a picture.



I don't understand where the numbers -0.67, 2.67, and 6 come from. It may be super obvious and I'm just missing it for some reason but if anyone could explain to me I'd really appreciate it! Thanks

 

[pka]

In my textbook, there's an example inequality problem that looks like this: -1/(x+5) ≥ -3x/(x+5), and the instructions simply say to solve. I understand that you have to get zero to one side, which gets you 0 ≥ (-3x+1)/(x-5), and then set the numerator and the denominator equal to zero to find the endpoints, x=1/3 and x=5.

\(\displaystyle \begin{align*}\frac{-1}{x+5}&\ge\frac{-3x}{x+5} \\\frac{3x}{x+5}+\frac{-1}{x+5}&\ge 0\\\frac{3x-1}{x+5}&\ge 0 \end{align*}\)
Now clearly there are two critical values: \(\displaystyle x=-5~\&~x=\tfrac{1}{3}\).
Two such values create three critical regions: \(\displaystyle (-\infty,-5),~\left(-5,\tfrac{1}{3}\right)~\&~\left[\tfrac{1}{3},\infty\right)\)
Because these are continuous, all we need it to test three representative values: lets say \(\displaystyle x=-6,~x=0,~\&~x=1\)
DO IT! Post your results. Tell us what they mean.

 

[Dr.Peterson]

In my textbook, there's an example inequality problem that looks like this: -1/(x+5) ≥ -3x/(x+5), and the instructions simply say to solve. I understand that you have to get zero to one side, which gets you 0 ≥ (-3x+1)/(x-5), and then set the numerator and the denominator equal to zero to find the endpoints, x=1/3 and x=5.

Now, the next part is what I don't understand. The example problem says that we get the intervals (-∞,1/3], [1/3,5), and (5,∞), but then says that we must check the intervals one by one to see which factors agree in sign. Here's a picture.

View attachment 11815

I don't understand where the numbers -0.67, 2.67, and 6 come from. It may be super obvious and I'm just missing it for some reason but if anyone could explain to me I'd really appreciate it! Thanks

No, it isn't super obvious. And no, you don't have to understand it.

There are a couple problems here. One is that you have apparently miscopied the problem when you first stated it, changing the sign in the denominator. That's why pka's work gives a different answer than what you show.

The other is that they used decimals, which is a REALLY BAD THING because rounding could lead to errors.

I suppose that they chose -2/3 and 2 2/3 as test points because they saw they'd be multiplying by 3; but then they wrote them as decimals and did something I can't even figure out to get an approximate fraction rather than an exact one. Or maybe they just wanted to show that it doesn't matter what numbers you choose, as long as they are in the right intervals. I'd believe that better if they explicitly stated that they had deliberately made a stupid choice. More likely, those made sense in some other problem, and they cut-and-pasted their work.

[Well, actually it looks like they subtracted 1 from 1/3 to get their first test point, then averaged 1/3 and 5 for the second, and added 1 to 5 to get the third. I can see no good reason to do that.]

I would always choose the easiest numbers possible (in this case, probably 0, 1, and 6); but I would emphasize that they are arbitrary, and you don't have to pick the very best possible numbers (or try to guess what your teacher would pick).

Also, when I do these, I recognize that I don't need exact calculations, just the signs. All that matters for x < 1/3 is that the numerator is positive and the denominator is negative, so the fraction is less than 0. Any effort used in finding the actual value is wasted.

 

[lolily]

Okay, I understand now. We just need to pick a random number within the interval to test. Thanks!