Number of integral points in the given enclosed region

Question

Questions:

How many points with integral coordinates lie inside the region bounded by the lines $|x| = 2$, $x + y = 12$ and $x – y = 5$?

My Working:

I drew the graphs and looked at the extreme points of the trapezium.

Then, I individually confirmed each integral coordinates whether they lie in the given enclosed region. I got $90$ such points.

Given answer:

$48$.

My Doubt:

Who is wrong and why is there such a difference of approximately 2 times between me and the answer key?

Also, is there any other (smarter) way than to individually calculate/identify the integral points and listing them?

Answer 1

$$\sum\limits_{x=0}^8 \left( 1 + \left\lfloor \frac{35 - 4x}{y} \right\rfloor \right) = 30$$

Answer 2

Divide the region $ABCD$ into into three regions : rectangle $AFED$ and two triangle $ABF$ and $DEC$.

No of points with integer coordinate inside the rectangle $AFED $ (including the points on $AF$ and $ED$ ) is $$14×5-2\times 14=42$$

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Let us consider the rectangle $DECG$.

No. of points with integer coordinate inside the rectangle $DECG$ is $$5\times 5-2\times (5+ 3)=9$$

Remove no.of diagonal entries with integer coordinate and divide by $2$

No. of points with integer coordinate inside the triangle $DEC$ is $$\frac{9-3}{2}=3$$

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Similarly no. of points with integer coordinate inside the triangle $ABF$ is $3$.

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Hence total number of points with integer coordinate inside the region $ABCD$ is $42+3+3=48$