Question

George tells you that when variables are in the denominator, the equation 4/5 + 3/x+1/2 becomes unsolvable. George explains, "There is a value for x that makes the denominator zero, and you can't divide by zero." Demonstrate to George how the equation is still solvable and explain your reasoning.

Answer 1

im a little upset just cos i know this example should be easier than it is but i can't figure out what to do. i know you have to find the LCD, which i guess would be 5? but i dunno. and also it's supposed to be

\[\frac{ 4 }{ 5 } + \frac{ 3 }{ x }=\frac{ 1 }{ 2 }\]

Answer 2

@whpalmer4 @tkhunny hey guys any of you guys free right now?

Answer 3

Thank you guys so much for viewing. I'm just a little confused on this one and would love some help

Answer 4

Your LCD would also have \(x\) in it...

Why not multiply all the way through by \(5*2*x\), which will "clear" all of the fractions?

Answer 5

ohhh ok

Answer 6

In this particular instance, the left side of your equation is defined for all x other than zero. Why not try the actual solution before attempting to answer the question posted in this problem? Follow whpalmer's suggestion.

Answer 7

so 10x?

Answer 8

Please show all your work at one time, not just tidbits.

Answer 9

^ i tried to do that

Answer 10

but my actual work doesnt make any sense

Answer 11

At least, identify what 10x is supposed to represent.
Yes, you could multiply all terms of the given equation by 10x, or by 2*5*x. Do this now, please, and share your result.

Answer 12

\[\frac{4}5*(5*2*x) + \frac{3}{x}*(5*2*x) = \frac{1}{2}*(5*2*x)\]

what do you get if you simplify that?

Answer 13

10x/2?

Answer 14

where are you getting a /2 from? what does each term simplify to?

Answer 15

let's find the mistake and fix it so you never do that again :-)

Answer 16

ok so you take the 5*2*x which equals 10x

Answer 17

Again, please show your entire result, not just part of it. whpalmer4 seems to be in agreement with me that we need to see what you are doing.

Answer 18

multiply it by the 1/2

Answer 19

and then you get 10x/2

Answer 20

i think

Answer 21

might be easier to just keep it like I wrote it and cancel out matching things in numerator and denominator first

Answer 22

Let's start over. Multiply EVERY term of the given equation by 10x. share ALL of the result.

Answer 23

umm ok lemme just look at it again

Answer 24

I'm going to let my esteemed colleague finish this off, you're in good hands!

Answer 25

ok lol this really doesnt make any sense to me

Answer 26

Explain what doesn't make sense. I need to know where you're coming from. We have an equation, above. I'm asking you to multiply each term of the equation by the LCD, which is 10x. Questions about that?

Answer 27

#1 Thing to do. Worry about the Domain.

1) What is that 'x' doing in the denominator?
2) We better make sure that thing never takes on the value zero (0).
3) \(Domain:\;x \ne 0\)
4) Whew! That was close. Now that we KNOW x cannot be zero, we can proceed without danger.

Answer 28

wait wait i think i got the answer. is it 10? if i write out my work will you guys tell me if im right?

Answer 29

Please show your work. Otherwise, no feedback from me. sorry, but that's the only way in which i could come to undrstand your thinking.

Answer 30

so after simplifying what whpalmer told me, i did this
\[\frac{ 10x }{ 5 } + \frac{ 30x }{ x }=\frac{ 10x }{ 2 }\]

Answer 31

And before you share your work, why not test your result yourself? Merely substitute 10 for x and determine whether or not the equation is true.

Answer 32

that all simplifies to \[2x + 30 = 5x\]

Answer 33

Going to solve that equation?

Answer 34

and then you get 10x just by subtracting an ddividing

Answer 35

isnt that solving it?

Answer 36

I'm still worried about the Domain.

If we start with this: 3 = 7
Do we believe that to be correct?
Let's multiply both sides by zero (0). 3(0) = 7(0) ==> 0 = 0. There you go. 3 is equal to 7.

Of course, that is ridiculous. BEFORE you multiply by x, you MUST state that x is NOT zero.

It's not optional.

Answer 37

sorry indeed to sound critical, but your "10x just by subtracting an ddividing" is incorrect. subtracting and dividing are NOT related in any way, other than that both are algebraic operations.

Please go back to 4/5 + 3/x = 1/2.

Substitute x=10. Is the equation now true?

Answer 38

ok ill check

Answer 39

nope its not

Answer 40

also @tkhunny thank you but im still confused

Answer 41

that means, unfortunately, that x=10 may not be a solution.

Looking at this same equation yet again, I see that we have a "1/2" on the right side. Any chance that the left side could be made = to 1/2?

Answer 42

could i just have 1 person at a time help me

Answer 43

umm

Answer 44

tk?

Answer 45

no no i want you to help me

Answer 46

but yea i dunno how to make the left side equal a half

Answer 47

like i tried to keep the 3/x on one side and have the 4/5 be on the 1/2 side

Answer 48

then i tried to isolate 3/x

Answer 49

but idk how to do that

Answer 50

Agreed. However, there are two different questions, here. One is the solution of the equation. The other is the answer to George. The second part is easy. Talk about the Domain and say, "Silly George!".

Answer 51

Here's what I see: If you substitute 10 for x in the equation just mentioned, we get
8/10 + 3/10 = 1/2. Not true.

However, if the left side were 8/10 - 3/10, what would that come out to?

Answer 52

it'd be a half

Answer 53

so you multiply 4/5 by 2?

Answer 54

where are you getting 3/10 though?

Answer 55

right: 1/2 on the left, 1/2 on the right. But we supposedly got x=10; that's not a solution.
We can't just arbitrarily state that "that 10 is negative." We'd have to get -10 algebraically, or it's "no soap."

Answer 56

by assuming that your answer, x=10, is correct.

Answer 57

m8 im confused. i understand where you got 8/10 but what are you doing to the 3/x to make it 3/10?

Answer 58

Then the2nd term is 3/10.

Answer 59

Here's what I'm seeing:\[\frac{ 8 }{ 10 }+\frac{ 3 }{ x }=\frac{ 1 }{ 2 }\]... for only one value of x. What is that value? Hint: It's not 10.

Answer 60

lol

Answer 61

i mean

Answer 62

could you subtract 8/10 and 1/2?

Answer 63

\[\frac{ 1 }{ 10 }+\frac{ 3 }{ x}=\frac{ 1 }{ 2 }\]

Answer 64

that'd get get you 3/10=3/x

Answer 65

We have already shown that if x=10, the above equation is FALSE. What other choice have you for x?

Answer 66

Hint: is 0.8 - 0.3 = to 0.5?

Answer 67

yea

Answer 68

x=10 is incorrect. Choose another value for x to try.

Note that we MUST have the same denominator in each fraction.

Answer 69

dude i know that x=10 is incorrect

Answer 70

but i don't even know what you're asking anymore at this point like ??? ok yes, x=10 is NOT the answer but i don't see how that's helping me get the right answer

Answer 71

the thing is, i don't know what value for x to try

Answer 72

Note that \[\frac{ 3 }{ -10 } \]could be re-written so that the denominator is +10.

Answer 73

Try re-writing it.

Answer 74

where are you getting 3/10 from

Answer 75

if you're still using the x=10 thing i really don't wanna use it, it's just confusing me

Answer 76

First, where did the x=10 come from?

Answer 77

holy crap dude i dont know??? i thought that was the wrong answer so we weren't supposed to use that

Answer 78

can you, or SOMEONE, for the love of God, just give me a straight forward answer of how im supposed to solve this algebraically because i've been here for 45 minutes and all that's happened is that i've become even more confused than before

Answer 79

brriiarr
wait wait i think i got the answer. is it 10? if i write out my work will you guys tell me if im right? So, that "x=10" came from you.

Answer 80

All right. x=10 is not correct. Try x=-10.

Answer 81

ok lemme plug it in

Answer 82

well you can't do that

Answer 83

cos the 10 and the -10 cancel out

Answer 84

Never said they should cancel out or that they don't cancel out. Try the following: substitute -10 for x in \[\frac{ 4 }{ 5 }+\frac{ 3 }{ x }=\frac{ 1}{ 2}\]

Answer 85

i know lol that's what i did?

Answer 86

Note that this substitution will result in \[\frac{ 4 }{ 5 }-\frac{ 3 }{ 10 }=\frac{ 1 }{ 2 }\]

Answer 87

Is that true or not?

Answer 88

thats what i got on my paper. so then i did the lcd thing and so 4/5 turns into 8/10 and you subtract it and get 5/10

Answer 89

and 5/10 is equal to 1/2, right?

Answer 90

Which means that ... ??

Answer 91

1/2=1/2

Answer 92

so the answer is 1/2?

Answer 93

wait no

Answer 94

Both whpalmer and tkhunny asked that we pay special attention to the domain of this function. I asked the same in another way: what value can x NOT have? and answered it: x cannot be zero. x can be -10, however.

Answer 95

You are not "finding an answer." You are, actually, determining whether or not the equation is true. Is it true? Is it false?

Answer 96

wait so the answer is /-10

Answer 97

*-10