Question 360·Hard·Algebra

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A point $(x, y)$ with both coordinates integers must satisfy the following system of inequalities:

\begin{cases} y \ge 2x - 6 \\ y \le -x + 2 \\ -1 \le x \le 5 \end{cases}

How many such integer-coordinate points $(x, y)$ satisfy the system?

A system of inequalities asks for points that satisfy every rule at once. When the question requires integer coordinates, graph the inequalities in Desmos to find the possible integer $x$ -values. For each one, count the integer $y$ -values between its lower and upper bounds, then add the counts. Desmos shades noninteger points too, so don't count the whole shaded region.

Hints

Hint 1
Graph the inequalities together. Their overlap contains the points that satisfy every condition, but only integer $x$ -values in that overlap can lead to an integer-coordinate point. Which vertical columns should you count?
Hint 2
For one integer $x$ , the two slanted lines give the lowest and highest allowed $y$ -values. Because the inequalities allow equality, both ends count. How many integers lie between those two bounds, including the ends?
Hint 3
Use a Desmos list of the possible $x$ -values to evaluate both $y$ -bounds in order. Count each column with upper bound minus lower bound plus $1$ , then add the column counts.

Step-by-step

Graph, then count by column
Step 1
Find the possible integer x-values
Type all four inequalities into Desmos, putting the two limits on $x$ on separate lines. The shaded regions overlap from $x=-1$ to a tip between $2$ and $3$ . Since $x$ must be an integer, only $x=-1,0,1,2$ can work.
Step 2
Evaluate the y-bounds for those columns
The lower bound is the smallest allowed $y$ , and the upper bound is the largest. Type $x_1=[-1,0,1,2]$ , then $a=2x_1-6$ and $b=-x_1+2$ . Desmos shows matching lists: $a=[-8,-6,-4,-2]$ and $b=[3,2,1,0]$ . The last entries say that at $x=2$ , $y$ runs from $-2$ through $0$ .
Step 3
Include both ends when counting y-values
At $x=2$ , the allowed integers are $-2,-1,0$ : three values, not two. Subtracting the bounds counts the gaps between them, so add $1$ to count the values themselves. For inclusive integer bounds, count upper minus lower plus $1$ . Here that means $b-a+1$ for each column.
Step 4
Count the points in each column
Type $b-a+1$ . Desmos shows $[12,9,6,3]$ , one count for each $x$ in the same order. These counts include points on the boundary lines, because $\ge$ and $\le$ both allow equality.
Step 5
Add the column counts
Type $\operatorname{total}(b-a+1)$ . Desmos prints $30$ . So 30 integer-coordinate points satisfy the whole system. Choice C.

Question 360·Hard·Algebra

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