Question

Select the correct answer. For which system of inequalities is (3,-7) a solution? A. x + y < -4 3x + 2y < -5 B. x + y ≤ -4 3x + 2y < -5 C. x + y < -4 3x + 2y ≤ -5 D. x + y ≤ -4 3x + 2y ≤ -5

Answer 1

D. x + y ≤ -4

3x + 2y ≤ -5

To see if (3, -7) satisfies both inequalities.

x + y ≤ -4:

Substitute x = 3 and y = -7 into the first inequality:

3 + (-7) ≤ -4

-4 ≤ -4

Since -4 is less than or equal to -4, the first inequality is true for the point (3, -7).

3x + 2y ≤ -5:

Substitute x = 3 and y = -7 into the second inequality:

3(3) + 2(-7) ≤ -5

9 - 14 ≤ -5

-5 ≤ -5

Since -5 is equal to -5, the second inequality is also true for the point (3, -7).

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Answer 2

Answer: Choice D

x + y ≤ -4 and 3x + 2y ≤ -5

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Step-by-step explanation:

The given point has coordinates of x = 3 and y = -7.

Let's plug those coordinates into the first inequality of choice A.

x+y < -4

3 + (-7) < -4

-4 < -4

The last statement is false. A number cannot be smaller than itself.

Since the last inequality is false, it causes the first to be false for those x,y values. Therefore, we rule out choice A.

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We'll do the same idea for the first inequality in choice B

x + y ≤ -4

3 + (-7) ≤ -4

-4 ≤ -4

This time we get a true statement at the end. The key difference is the "or equal to" portion.

Let's check the other inequality of choice B

3x + 2y < -5

3(3) + 2(-7) < -5

9 - 14 < -5

-5 < -5

We run into a similar issue as we did with choice A. We have no choice but to cross choice B off the list as well.

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Choice C is eliminated for the same reason choice A was.

Choice D is the final answer because both inequalities involve "or equal to". So the -5 < -5 is now -5 ≤ -5 which is true.

It turns out that (3, -7) is on the boundary of each shaded region. It's at the intersection of the two boundary lines x+y = -4 and 3x+2y = -5.