Find the number of integer solutions

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Find the number of integer solutions

Mathematics
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\(\large \color{black}{\begin{align} y\leq 5-|x-1|,\ \ y\geq 0\hspace{.33em}\\~\\ \end{align}}\)
|dw:1434213766900:dw|
looks like 36 point

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Yes! Notice that number of points above a point on \(x\) axis, \( k\) is given by: \[5-|x-1| +1\]
so the number of lattice points is \[\sum\limits_{x=-4}^{6}6-|x-1|\]
\[\begin{align} &=1+2+3+4+5+6+5+4+3+2+1\\ &=36 \end{align}\]
how did u find interval \([-4,6]\)
-4, 6 are the x intercepts of \(f(x) = 5-|x-1|\)
we want \(y\ge 0\), so we're finding x intercepts
y did u take here 6, \(\sum\limits_{x=-4}^{6}\fbox{6} -|x-1|\)
|dw:1434214973352:dw|
whats the value of f(x) at x=2 ?
4
and how many lattice points are there above x=2 ?
5
so can we say the number of lattice points above x=2 is given by \(f(x) + 1\) ?
yes
\(f(x)+1 = ?\)
x+1
how ?
sry 6-|x+1|
Yep, that expression gives the number of grid points above \(x=k\) iterating it from x=-4 to x=10 gives the required answer
what is the value of this \(\large \color{black}{\begin{align} \sum\limits_{x=1}^{n} |x|\hspace{.33em}\\~\\ \end{align}}\)
|x| = x because all the numbers that x takes are positive
\[\large \begin{align} \sum\limits_{x=1}^{n} |x|&=|1|+|2|+|3|+\cdots +|n|\\~\\ &=1+2+3+\cdots +n\\~\\ &=n(n+1)/2 \end{align}\]
ok thnx

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