Question

Guys this one has me stumped- I need some serious help... I think I got it right, but a second look would be appreciate. Thanks!

Solve the following inequality
(-2)/(5-x)>= (1)/(x-2)

Write your answer as an interval or union of intervals
If there is no real solution, write “no solution”

Answer: [-1,2] U (5,infinity)

Answer 1

I'll give a hint on the first step:
\[-2(x-2) \ge 5 - x\]
Now solve for x

Answer 2

Is it (-1<=x<2) U (5, infinity)?

Answer 3

Ravina gave a very nice hint.

Answer 4

I'm lost completely

Answer 5

x<=-1 maybe???

Answer 6

That hint above is not going to help you. I'll write up something here that will help. Hold on.

Answer 7

ok :)

Answer 8

The reason that the above hint is faulty is that we are possibly multiplying (cross-multiplying) by a negative number and migh be encountering a sign change. The following avoids all that:

First, take that left side and multiply the top and bottom by -1. Since this -1/-1 is stil +1, we can always do that and not have to worry about a sign change:

(+2)/(x-5)>= (1)/(x-2)

That's the biggest key. Now, we can get the reciprocal of both sides and NOW do a forced sign change:

(x-5)/2 <= x-2

Multiply both sides by 2:

x - 5 <= 2x - 4

Subtract x from both sides (no sign change)

-5 <= x - 4

-1 <= x

See how this answer gives a different answer from what the other helper's wrong hint would give?

Answer 9

I see how we didn't switch the signs at the end. THAT's what I did wrong!

Answer 10

Yes, but the first person's hint was leading you down a path where there was an assumed "no sign change", when in effect, the solution would really have forced an immediate sign change, but one would have had to know the solution to know to change the sign! Very hard! Knowing the solution before finding the solution!

Answer 11

LOL! I've been working on this one question ALL day long!

Answer 12

Inequalities are often tricky. If you stay with safe operations, you can often get past whether to change the sign or not. Here, I multiplied the left side by +1 (-1/-1), so no sign change and that simplified things little. The second trick was the reciprocals and the sign change. That was key.

Answer 13

* "a little" not "little"

Answer 14

If I'm reading this correctly, there is no union of intervals at all! So the interval notation would be [-1,infinity). SMH- why oh why won't this sink in?!?!? #pounding head on wall

Answer 15

First off, where did you get that suggested or possible answer in the problem statement? Was that your work, or someone else's, or stated that was as a given answer?

Answer 16

That was my work. I took both equations on either side of the inequality to put 0 on the right. \[\frac{ x+1 }{ (x-5)(x-2) }\ge 0\]

Answer 17

My result was [-1,2), U (5, infinity). But I wasn't 100% sure...

Answer 18

There are other ways to work the problem than the one I gave you. But I concentrated on that one because it was probably the most pure and straight-forward from a mathematical point of view. Another way, without going into too much detail, unless you want to see it, and a method I sometimes use, is to drop the inequality and work with the equality. Then worry about the sign later. If you get an answer that is "linear", that is, "x" to the "1" power, you can just plug in x=0 to check which way to go with the sign. Sometimes that is faster.

Answer 19

I am absolutely noo good at math- I'm 36 it's been a while since I've had Algebra so I'm scouring you tube for tutorials. I have 14 questions on this test and I'm done with college for good...

Answer 20

So what you just wrote looks like this to me \[\alpha \eta \theta \iota \xi \xi \Delta \Theta \Lambda= pancakes bc aliens live on mars\]

Answer 21

lol - I will NEVER do algebra after this test!!

Answer 22

I have a lot of sympathy with anyone who has a hard time with math. I really want to help them, if for no other reason, than to keep them from going through a hard time. Math comes easy for me, but other things are hard for me. I'm a slow reader and that bugs me. So, if I can help, I do. That method of putting everything over to the left can get you in trouble because we still don't know if x is positive, negative or could be both.

That's ok about never doing math again, but it is often very useful. Math is like a friend that speaks a foreign language. A friend that can be very helpful.

Answer 23

i know but that method is all that I've ever learned. I've always been left brained so the creative stuff has always been easy for me. This stuff kills me, it's like trying to climb a rope made of water for me. Lol.

Answer 24

Let me see if I can solve it that way for you if it helps at all.

Answer 25

I don't mean to waste your time tcarroll. I sooooo appreciate your help very very much!

Answer 26

It's up to you. I try to help people learn the way they are accustomed to. You can stick around if you like, because I'm going to do it anyway. But you don't have to stick around if you don't want to. np.

Answer 27

No I'm here, and I would love to know

Answer 28

ok, I started trying to do it that was and I instinctively saw that this over-complicating the issue for 2 reasons. One, it introduces a quadratic in the denominator and two, you can't do any cross-multiplying with the zero on the right side. At least the first helper, though she had a wrong sign handling, was doing a good thing with cross-multiplying with non-zero numbers. She also did a good thing with staying away from introducing a quadratic unnecesaarily. But she didn't quite understand the sign handling. If she dropped the inequality and worried about it later, that would have been fine and that was very close to my alternative method. But don't go the route you went, that makes it much harder. I just instinctively stay away from approaches like that because it makes the problem so much harder instead of easier.

Answer 29

If I work it this way and solve the inequality- reversing the signs like you said- I STILL come out with x <= -1

Answer 30

I took −2(x−2)≥5−x which is what Travina had hinted and worked out the x and switched the signs.

Answer 31

One way to help yourself from getting lost is to take your solution and plug it back into your inequality and see if it works. If you went with x <= -1, then x=0 would not be a solution. If you put x=0 (I chose that because it's a simple value) into the original inequality, you would see that it DOES work. So, x <= -1 can't be the solution. It' s the othe way around.

Answer 32

AHHHH!!! So -1<= x and to check if x=0 then that would be a true statement!

Answer 33

I hope you get paid for helping people like me! lol

Answer 34

Yes, and that would mirror my alternative method instead of the first method I showed. I really suggest going over my post #2 where I went from step to step. If you can follow it, it's all there. This idea of "seeing which side of -1 we are on" is "worrying about the sign last" and is ok, but is actually a short-cut. My first method illustrates sound mathematical laws and is actually the better way of the two.

Answer 35

I get paid in satisfaction that I'm alleviating suffering. I say that both in fun and seriously.

Answer 36

btw, I'm quite a bit older than you and I know how it can be difficult to suffer through things like this, so I can commiserate.

Answer 37

I can't believe you actually enjoy math- that's sadistic (no offense)
So my answer for my problem would be [-1,infinity)

Answer 38

I'm doing this in between changing diapers and picking kids up from school. It's tough :/

Answer 39

Yes, you have the correct representation of the answer with that. It's funny that everyone in my family thinks I'm nuts for liking math. You should just see the things I do for fun! People just shake their heads!

Answer 40

You are awesome, and I appreciate you spending the time teaching me this stuff! you have no idea :)

Answer 41

Here's something I did for fun yesterday: I came up with a computer algorithm for computing the interest rate for a series of annuity payments, given the amount of the payments and the future value of the total sum. Yes, I'm certifiable!

Answer 42

That is certifiable! I'm married to an accountant, and she likes doing stuff like that, but with tax season here, the in house tutor is gone.

Answer 43

Well, you're a nice person, and I very much enjoyed helping. I just hope i actually did help. But it's all there in my second post. When you're done with diapers, please go over it, you just might benefit from it.

Answer 44

Ditto Carroll - it was a pleasure! Thank you

Answer 45

You, too! Bye for now, thx for the recognition! Good luck in all of your studies! @Jfcarlucci

Answer 46

Are you still there? I have something important to add to the problem! My answer is NOT complete! I figured out the true complete answer! But I have to tell you about it! @Jfcarlucci

Answer 47

I looked at my solution and it was bothering me because it left something out. The answer is not:

-1 <= x

It is:

-1 <= x < 2 union with 5 < x

[-1, 2) U (5, infinity)

So, you were right! I neglected to go back to the original inequality. Oh, I feel so stupid for having neglected that and I'm terribly sorry that I didn't go through that with you. I can't tell you how sorry I am. Here's a summary of how I (finally) did it. You came up with the correct answer yourself, so you just might want to stick with your own method, but what I did was, first I ignored the inequality and stuck with the equality. I cross-multiplied and came up with x = -1. I went back to the original inequality and saw that the x-number line is broken up into the areas to the left of -1, the area between -1 and 2, the area between 2 and 5, and the area to right of 5. Four areas. I then examined a value from each area to see which areas held true or false for the inequality. I'm so sorry, I truly am. It's terrible that I didn't see that right off and misled you. I'm sorry. But I do now have the true answer and it matches yours. Here's an attached graph and the red graph is -2/(5 - x) and you can see the areas. Take care of yourself.