## mathmath333 one year ago Find the number of integer solutions

1. mathmath333

\large \color{black}{\begin{align} y\leq 5-|x-1|,\ \ y\geq 0\hspace{.33em}\\~\\ \end{align}}

2. ganeshie8

|dw:1434213766900:dw|

3. mathmath333

looks like 36 point

4. ganeshie8

Yes! Notice that number of points above a point on $$x$$ axis, $$k$$ is given by: $5-|x-1| +1$

5. ganeshie8

so the number of lattice points is $\sum\limits_{x=-4}^{6}6-|x-1|$

6. ganeshie8

\begin{align} &=1+2+3+4+5+6+5+4+3+2+1\\ &=36 \end{align}

7. mathmath333

how did u find interval $$[-4,6]$$

8. ganeshie8

-4, 6 are the x intercepts of $$f(x) = 5-|x-1|$$

9. ganeshie8

we want $$y\ge 0$$, so we're finding x intercepts

10. mathmath333

y did u take here 6, $$\sum\limits_{x=-4}^{6}\fbox{6} -|x-1|$$

11. ganeshie8

|dw:1434214973352:dw|

12. ganeshie8

whats the value of f(x) at x=2 ?

13. mathmath333

4

14. ganeshie8

and how many lattice points are there above x=2 ?

15. mathmath333

5

16. ganeshie8

so can we say the number of lattice points above x=2 is given by $$f(x) + 1$$ ?

17. mathmath333

yes

18. ganeshie8

$$f(x)+1 = ?$$

19. mathmath333

x+1

20. ganeshie8

how ?

21. mathmath333

sry 6-|x+1|

22. ganeshie8

Yep, that expression gives the number of grid points above $$x=k$$ iterating it from x=-4 to x=10 gives the required answer

23. mathmath333

what is the value of this \large \color{black}{\begin{align} \sum\limits_{x=1}^{n} |x|\hspace{.33em}\\~\\ \end{align}}

24. ganeshie8

|x| = x because all the numbers that x takes are positive

25. ganeshie8

\large \begin{align} \sum\limits_{x=1}^{n} |x|&=|1|+|2|+|3|+\cdots +|n|\\~\\ &=1+2+3+\cdots +n\\~\\ &=n(n+1)/2 \end{align}

26. mathmath333

ok thnx