Pick an answer choice and plug its $x$-value into each right-hand side ($-2x+4$ and $x-1$), then compare the result with the given $y$-value using the inequality sign.

Remember that $>$ does not allow equality, while $\le$ does. A point exactly on the line for $y = -2x+4$ will not satisfy $y > -2x+4$.

Check each ordered pair against both inequalities. As soon as one inequality fails, you can eliminate that choice and move on.

In Desmos, enter the two inequalities on separate lines: y > -2x + 4 and y <= x - 1. Desmos will shade the region that satisfies each inequality; the solution set is where the shadings overlap.

Enter each option as a point on separate lines, for example (2,0), (1,0), (3,1), (2,2). Visually check for each point whether it lies inside the overlapping shaded region that represents solutions to both inequalities (and not just on a boundary excluded by >). The point that lies in this overlapping region is the solution.

The system is:

An ordered pair $(x, y)$ satisfies this system only if, when you plug in its $x$ and $y$ values, both inequalities are true at the same time. If even one inequality is false, that ordered pair is not a solution.

Take a generic point $(x, y)$.

• In the first inequality $y > -2x + 4$, substitute the $x$-value into $-2x+4$, compute the result, and compare it with the $y$-value. The $y$-value must be greater than that result.
• In the second inequality $y \le x - 1$, substitute the same $x$-value into $x-1$, compute it, and check that the $y$-value is less than or equal to that result.

We will apply this procedure to the answer choices one by one.

Choice A: $(2, 0)$

• First inequality: $y > -2x+4$ becomes $0 > -2(2)+4 = 0$. This says $0>0$, which is false because $>$ does not allow equality. So $(2, 0)$ does not satisfy the system.

Choice B: $(1, 0)$

• First inequality: $y > -2x+4$ becomes $0 > -2(1)+4 = 2$. This says $0>2$, which is clearly false. So $(1, 0)$ does not satisfy the system.

Choice D: $(2, 2)$

• First inequality: $y > -2x+4$ becomes $2 > -2(2)+4 = 0$. This says $2>0$, which is true.
• Second inequality: $y \le x-1$ becomes $2 \le 2-1 = 1$. This says $2\le1$, which is false. So $(2, 2)$ does not satisfy the system either.

Only one choice remains to test.

Choice C: $(3, 1)$

• First inequality: $y > -2x+4$ becomes $1 > -2(3)+4 = -6+4 = -2$. This is $1>-2$, which is true.
• Second inequality: $y \le x-1$ becomes $1 \le 3-1 = 2$. This is $1\le2$, which is also true.

Since $(3, 1)$ makes both inequalities true, it is the ordered pair that satisfies the system.

Answer: $(3, 1)$.