Dear feanur,

I have troubles understanding absolute values. Quite frankly, I never thought more of it than "Just turn any negative number into that number without the negative sign", but here was an exercise that caught me by surprise.

A is the set {−100,−99,…,−1,0,1,…,99,100}. The set A includes precisely all the numbers -100 to 100 (including -100 and 100).
How many collections of coordinates (x,y) with x∈A and y∈A meet the following equation?
|x|=|y|+1

I had no idea where to start, so I started off with a cheat by plotting this only to be in awe of the unexpected result. The only thing it tells me really, is that for each value y there will be two answers x, but I don't know how to shape that into an equation so I can work with it.

Can you help me?

Rik

(80*4)+2

[(2,1), (3,2),...] and all their combinations in interval [-100,100] are all of the solutions (except the zero case), so four combinations for each set of coordinates ++ +- -+ -- with (1,0) and (-1,0) as the remaining coordinates

This post was edited by Zekdawg on May 17 2017 03:14pm

If you want an equation for a solution over the reals:


y= +-(x-1) , 1 <x <=100 and x=+-1 , y=0


Also correction to my first post it should be
(99*4)+2

This post was edited by Zekdawg on May 17 2017 03:42pm

I'm no feanur but I believe if you use the piecewise definition you will have plenty to work with


/e

The piecewise seems to be key here

I count 6 cases to consider

This post was edited by brigadier on May 17 2017 11:45pm

I cheated even further to find that the answer is 398 (so 99*4+2 is a correct solution)
Can you describe to me the theory behind piecewise definitions and/or the steps of what Zekdawg has done? I can't follow

Take a moment and graph

Y=|x|

What shape does the graph make?



Correct a "v" shape


Now take a moment and think hiw can you model the right hand side of the graph using a linear equation


Great job y = x


Is this for all x?


Yes it's only the positive x
(X>0)

Meow try and model the left hand side



Yes y=-x would make that line


Again is this for all x?



Good work it is only the negative side (x<0)



Meow have the inequalities you wrote describe all x values?


We were missing x=0!!



Need to describe the positive version and negative version of the graph normally it is broken up into two lines but in your case I believe t would be 4. Then we have to consider the pointy part of the graph

I have to head to work

Take a moment view the graph again

Model each linear you see using y=mx+b restrict domains as needed

So I guess this means that, |y|=|x|+1 mean that y = x+1 , y = -(x+1) = -x-1?

This makes sense, |y| = |x|+a is the sum of y = |x|+a , y = -(|x|+a) on graph apparently. I still can't quite translate it to the answer of 398, but at least it helps me visualize things a little bit.

break it up into the 4 lines then think about the possible solutiosn for each of the 4 lines

Didn't want to open a new thread haha
I once already got an answer for this very question, however, I forgot how to solve it. Besides, somebody gave me one explanation that I can't keep up with but he hasn't been back on forum since giving the explanation.

f(x) = x^3 - 11x^2 - 25x - 13
y = p*x + q
Line Y is the tangent of f(x) at A(a,f(a)) and intersects with f(x) at B (13,0). What is the sum of p+q?





I don't understand just how to factorize this problem. Can anybody tell me what tool I need to do this? I'll be happy to look it up myself from there.

/e Is there a way for me to assume that A and B are the only solutions? If so, one can assume that there has to be a factor (x-13).

This post was edited by Forg0tten on May 19 2017 03:10am