Which statement is correct about the system of inequalities x+2/y < 1 and y−7 > 5?

Question

1. The set of all ordered pairs whose x-coordinate is in the interval (10, ∞)  and whose y-coordinate is in (12, ∞) is the complete solution for the system.
2. The set of all ordered pairs whose x-coordinate is in the interval (-∞, 1) and whose y-coordinate is in (5, ∞) is the complete solution for the system.
3. The set of all ordered pairs whose x-coordinate is in the interval (-∞, 10) and whose y-coordinate is in (5, ∞) is part of the complete solution for the system.
4. The set of all ordered pairs whose x-coordinate is in the interval (-∞, 10) and whose y-coordinate is in (12, ∞) is part of the complete solution for the system.

Answer 1

The second inequality is the easier one to work with. What statement can you reach about y?

If y - 7 > 5, then we add 7 to both sides and arrive at y > 12. Thus, y is in (12, ∞). This fact narrows down our possibilities to the first or the fourth option.

To narrow it down further, we need to consider the first inequality. I am going to assume that you left out parentheses such that the inequality should read (x + 2)/y < 1. Multiply both sides of the inequality by y such that

x + 2 < y (Note that we can do this because we know that y > 0)

But we know that y < 12, so if x + 2 < y and y < 12, x + 2 < 12 must be true by the transitive property of inequalities. Thus, x < 10 so we say that x is in (-∞, 10).

Since x is in (-∞, 10) and y is in (12, ∞), the fourth option is the correct one.