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the difference between twice a number and 8 is twenty more than nine times the number. find the number

 Jul 29, 2014

Best Answer 

 #9
avatar+97586 
+13

Thank you Anonymous for pointing that out  

I think that you each have half the solution.

The difference between 5 and 3 =2

The difference between 3 and 5 is also 2

So the difference between 2 numbers n1 and n2 is  |n1-n2|

So the problems becomes

|2x-8| = 9X+20

This has 2 solutions

$$\begin{array}{rllrrll}
2x-8&=&9x+20 &or& \qquad 2x-8&=&-(9x+20)\\\\
-7x&=&28 &or& 2x-8&=&-9x-20\\\\
x&=&-4 &or& 11x&=&-12\\\\
x&=&-4 &or& x&=&\frac{-12}{11}\\\\


\end{array}$$

.
 Jul 30, 2014
 #1
avatar+3450 
+13

We just need to "translate" this to algebraic language and solve! (just need to set it up as an algebra problem)

Let's bold the key words here:

"the difference between twice a number and 8 is twenty more than nine times the number"

"difference" means subtract

"twice a number" can be written as 2N

"is" means equals or =

"more" means to add

"nine times the number" can be written as 9N

 

 

Let's write it out and solve for N:

2N-8=20+9N          ---Subtract 2N from both sides

-8=20+7N             ---Subtract 20 from both sides

-28=7N                ---Divide both sides by 7

-4 = N

so

N = -4

 Jul 29, 2014
 #2
avatar+97586 
+3

Another excellent answer Ninja!

 Jul 29, 2014
 #3
avatar+96302 
+3

I think we can retire now, Melody!!!

It looks as if ND can handle the forum by himself............!!!!

 

  

 Jul 29, 2014
 #4
avatar+3450 
+8

Thanks guys.

I wouldn't say that I could handle this forum by myself though. You guys do alot around here!

 Jul 29, 2014
 #5
avatar+97586 
+3

Between you and Aziz, I think Chris and I can put our feet on the table and enjoy our coffee and donuts without a concern in all the world.  Does that sound good Chris?

 Jul 29, 2014
 #6
avatar+96302 
+3

Excellent idea.....at the end of the week.....if we deem that ND and Aziz have worked hard enough......we might let them share the coffee and donuts with us......but only if they work hard enough.... and we'll be the final judges of that....I may give up my sunglasses easily, but I don't give up coffee and donuts so easily..!!!!!

 

  

 Jul 29, 2014
 #7
avatar+97586 
+3

NO NEITHER DO i!!

 Jul 30, 2014
 #8
avatar
+8

What is the "difference" between -8 and +8 ?

ND says that "difference" means subtract, but does that mean I calculate 8 - (-8) = 16, or do I calculate (-8) - 8 = -16 ? Is the "difference" 16 or -16 ?

Does it matter which number comes first ? That is, if I ask for the "difference" between -8 and +8, would I get a different answer than if I ask for the "difference" between +8 and -8 ? Shouldn't they be the same ?

I think that the "difference" between two numbers is the modulus of one number minus the other, that is, the "difference" between two numbers is a positive number. The "difference" between -8 and +8 is +16, not -16. They are 16 apart on the number line.

That is why I think that ND's answer to this question is wrong. ND's "difference" 2N-8 implies that 2N is greater than 8. Then, if the algebra says that N=-4, (producing a "difference" of -16), we have a contradiction, meaning that there is no N such that 2N>8 meeting the requirements of the question.

If there is no solution for which 2N>8, we have to consider the possibility that 2N<8, (and, in passing, the possibility that 2N=8, easy to reject that).

2N<8 leads to the equation 8 - 2N = 9N + 20, and the solution to that is N = -12/11.

 Jul 30, 2014
 #9
avatar+97586 
+13
Best Answer

Thank you Anonymous for pointing that out  

I think that you each have half the solution.

The difference between 5 and 3 =2

The difference between 3 and 5 is also 2

So the difference between 2 numbers n1 and n2 is  |n1-n2|

So the problems becomes

|2x-8| = 9X+20

This has 2 solutions

$$\begin{array}{rllrrll}
2x-8&=&9x+20 &or& \qquad 2x-8&=&-(9x+20)\\\\
-7x&=&28 &or& 2x-8&=&-9x-20\\\\
x&=&-4 &or& 11x&=&-12\\\\
x&=&-4 &or& x&=&\frac{-12}{11}\\\\


\end{array}$$

Melody Jul 30, 2014
 #10
avatar+96302 
+3

I agree with Melody's answer......however......I'll bet that 99% of us would have worked it just like ND did.......(I know I would have !!!)

 

 

 Jul 30, 2014
 #11
avatar
+8

Suppose that the number is -4.

What is twice that number ? Answer -8.

What is the difference between -8 and 8 ? Answer 16.

This is 20 more than 9 times the number, so what is 9 times the number ? Answer -4.

So 9 times the number is -4 ?

Wrong, 9 times the number is -36.

The difference between twice the number and 8 can be written (should be written) as    |2x -  8| and the equation will be |2x - 8| = 9x + 20, (as you say).

If the expression (2x - 8) is positive, the equation can be written as 2x - 8 = 9x + 20. However, this leads to a contradiction, since x = -4 makes the expression negative.

Look at it this way. Substitute your x = -4 into the equation |2x - 8| = 9x + 20 and see what you get. |2(-4) - 8| = 9(-4) + 20, |-16| = -36 + 20, 16 = -16. It doesn't work,      x = -4 is not a solution of the equation.

 Jul 30, 2014
 #12
avatar+97586 
+8

Anonymous is correct!

I did it the right way but I forgot that the answers to these types of questions always have to be checked.

and x=-4 simply does not work!

Taking my own little equation

|2x-8| = 9X+20

LHS = |2*-4-8|=16

RHS = 9*-4+20 = -16  not equal LHS

So as Anonmous has so adamantly stated, the ONLY answer is x=-12/11

THANK YOU ANONYMOUS!

Note: I should have checked x=-12/11 too, but it will be correct.

Does this answer satisfy you Anonymous.  If not please say so - we are all here to learn.

 Jul 30, 2014
 #13
avatar
+8

Yes, that's fine. Pleased to be able to help.

 Jul 30, 2014
 #14
avatar+97586 
+3

I also really liked the fact that this was a serious thread and that no one posted any irrelevant, distracting posts in the middle of it.  Thank you everyone.

 Jul 30, 2014
 #15
avatar+3450 
+3

Thanks anonymous.

If we're not carefull, be can just solve equations like these in a breeze.

Thanks for the correction, I didn't even think about that.

 Jul 30, 2014
 #16
avatar+97586 
+3

Continued here

http://web2.0calc.com/questions/another-look-at-something

Why?  I don't know - ask CPhill.

 Jul 31, 2014

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