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

Guest Jul 29, 2014

#9**+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}$$

Melody
Jul 30, 2014

#1**+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**

NinjaDevo
Jul 29, 2014

#3**+3 **

I think we can retire now, Melody!!!

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

CPhill
Jul 29, 2014

#4**+8 **

Thanks guys.

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

NinjaDevo
Jul 29, 2014

#5**+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?

Melody
Jul 29, 2014

#6**+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 tha*t....I may give up my sunglasses easily, but I don't give up coffee and donuts so easily..!!!!!

CPhill
Jul 29, 2014

#8**+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.

Guest Jul 30, 2014

#9**+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**+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 !!!)

CPhill
Jul 30, 2014

#11**+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.

Guest Jul 30, 2014

#12**+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.

Melody
Jul 30, 2014

#14**+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.

Melody
Jul 30, 2014

#15**+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.

NinjaDevo
Jul 30, 2014

#16**+3 **

Continued here

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

Why? I don't know - ask CPhill.

Melody
Jul 31, 2014