Is (3, -7) A Solution? Test Your Inequality Skills

Hey guys! Ever stared at a system of inequalities and wondered if a specific point is actually part of the solution? Today, we're diving deep into a classic math problem that'll sharpen your skills: figuring out which system of inequalities has $(3,-7)$ as a solution. This ain't just about plugging in numbers; it's about understanding how inequalities define regions and how a single point fits (or doesn't fit!) into those regions. We'll break down the process step-by-step, so by the end of this, you'll be a pro at testing points and can confidently tackle any similar problems that come your way. Let's get started and see if we can find the right fit for our point $(3,-7)$!

Understanding Systems of Inequalities

Alright, let's talk about what a system of inequalities actually is, guys. Think of it like a set of rules that a bunch of numbers has to follow at the same time. When we're dealing with inequalities like $x+y < -4$ or $3x+2y < -5$, we're not looking for exact values that make the equation true (like in regular equations), but rather a range of values. For a system of inequalities, a solution is any point $(x, y)$ that makes every single inequality in the system true simultaneously. It's like trying to find a spot on a map where several different shaded areas overlap. The point $(3, -7)$ has an $x$-coordinate of 3 and a $y$-coordinate of -7. Our mission, should we choose to accept it, is to plug these values into the inequalities provided in each option and see which set of inequalities both hold true for $(3,-7)$. This is a crucial skill in algebra, especially when you move on to graphing these solutions, because each inequality represents a region, and the solution to the system is the intersection of all those regions. So, for each option, we're going to perform two checks. If even one check fails for an option, that option is out. We need a perfect score of two-for-two for the correct answer. This process is fundamental to understanding graphical solutions and constraint satisfaction problems in more advanced mathematics and even computer science. So, let's buckle up and start testing!

Testing the Point $(3,-7)$

Now for the main event, guys: testing our point $(3,-7)$ against the provided systems. Remember, $(x,y) = (3,-7)$, so $x=3$ and $y=-7$. We need to substitute these values into each inequality in each option and verify if the statement is true. Let's go through them one by one.

Option A

Our first inequality is $x+y < -4$. Plugging in our values, we get $3 + (-7) < -4$. This simplifies to $-4 < -4$. Is this true? Nope! $-4$ is not less than $-4$; it's equal to $-4$. Since the first inequality is false, the entire system A cannot be the correct solution, regardless of the second inequality. We can stop here for Option A. Important note, guys: pay close attention to the inequality signs – less than ($<$) is different from less than or equal to ($\leq$).

Option B

Option B has two inequalities: $x+y \leq -4$ and $3x+2y < -5$. Let's test the first one: $x+y \leq -4$. Substituting our values, we get $3 + (-7) \leq -4$, which simplifies to $-4 \leq -4$. Is this true? Yes! $-4$ is less than or equal to $-4$ because it's equal. Now, let's check the second inequality: $3x+2y < -5$. Plugging in $x=3$ and $y=-7$, we get $3(3) + 2(-7) < -5$. This becomes $9 + (-14) < -5$, which simplifies to $-5 < -5$. Is this true? Again, nope! $-5$ is not less than $-5$. Since the second inequality is false, Option B is also not the correct system. We needed both to be true, and the second one failed.

Option C

We're on to Option C, which includes $x+y < -4$ and $3x+2y < -5$. Let's test the first inequality: $x+y < -4$. Substituting $x=3$ and $y=-7$, we get $3 + (-7) < -4$. This simplifies to $-4 < -4$. As we saw in Option A, this is false. So, Option C is also incorrect. It seems we might have a typo in the question or options provided, as none of the options seem to work perfectly with the strict inequality signs. However, if we assume there might be a slight adjustment needed or if the original problem had different options, let's re-evaluate with a hypothetical scenario or consider the closest fit.

Self-correction: Let's re-read the prompt and options carefully. It's possible I missed something or there's a subtle point. The initial assessment of A and C failing on the first inequality $x+y < -4$ because $3+(-7)=-4$ is correct. For B, the first inequality $x+y \leq -4$ is true since $-4 \leq -4$. However, the second inequality $3x+2y < -5$ resulted in $-5 < -5$, which is false. This means none of the options provided exactly fit the criteria for $(3,-7)$ to be a solution if all inequalities are strictly followed as written.

Let's assume there might be a typo in the question or options and proceed as if one of them should be the answer, looking for the closest match or re-examining the math.

Revisiting Option C:

• First inequality: $x+y < -4$. Substituting $(3,-7)$: $3 + (-7) < -4 \implies -4 < -4$. This is False. So C is out.

Revisiting Option B:

• First inequality: $x+y \leq -4$. Substituting $(3,-7)$: $3 + (-7) \leq -4 \implies -4 \leq -4$. This is True. Okay, this one works!
• Second inequality: $3x+2y < -5$. Substituting $(3,-7)$: $3(3) + 2(-7) < -5 \implies 9 - 14 < -5 \implies -5 < -5$. This is False. So B is out.

Revisiting Option A:

• First inequality: $x+y < -4$. Substituting $(3,-7)$: $3 + (-7) < -4 \implies -4 < -4$. This is False. So A is out.

Conclusion based on strict interpretation: As it stands, none of the provided options A, B, or C have $(3,-7)$ as a solution because in each case, at least one inequality is not satisfied. This can happen sometimes with textbook problems or quiz questions; they might have a typo.

However, if we were forced to choose the