Solving Systems Of Inequalities

Dec 15, 2025 by Andrew McMorgan 32 views

Solving Systems of Inequalities: Finding the Right Point

Hey guys! Ever stared at a system of inequalities and wondered which point actually works? It's like a secret handshake for math problems. Today, we're diving deep into a common scenario, figuring out how to pinpoint the solution to a system. We'll break down this specific problem and make sure you're totally confident when you see one of these pop up. Our focus? The system:

\left\{\begin{array}{c} y>x-3 \ y \leq-2 x+5 \end{array}\right.

And we've got a few potential points to check: A. $(-3,-5)$ , B. $(3,-5)$ , C. $(5,-3)$ , and D. $(5,3)$ . Let's get this sorted!

Understanding the Goal: What Does a Solution Mean?

So, what are we actually looking for when we talk about a "solution to a system of inequalities"? Think of it this way: each inequality represents a region on a graph. For instance, $y > x - 3$ is everything above the line $y = x - 3$ , and $y \leq -2x + 5$ is everything on or below the line $y = -2x + 5$ . A solution to the system is any point $(x, y)$ that satisfies both inequalities simultaneously. It's the sweet spot where the shaded regions of both inequalities overlap. When you're given specific points like the options A, B, C, and D, your job is to test each point in both inequalities. If a point makes both inequalities true, then congratulations, you've found a solution! If it fails even one, it's out of the running for being the solution.

This is a fundamental concept in algebra and pre-calculus. Graphing these inequalities helps visualize the solution set as a specific area. The boundary lines might be dashed (for $>$ or $<$ ) or solid (for $\geq$ or $\leq$ ). In our case, $y > x - 3$ has a dashed boundary, meaning points on the line don't count for this inequality, while $y \leq -2x + 5$ has a solid boundary, meaning points on this line do count. The overlapping region is where all the solutions lie. When checking specific points, we're essentially seeing if those coordinates fall within that common overlapping area. It's a direct way to verify if a proposed point is part of the solution set, which is particularly useful in multiple-choice questions or when you need to confirm a specific value.

Remember, there isn't just one solution to a system of inequalities; there's usually an infinite number of them, forming a region. The question asks which of the given points is a solution, meaning we just need to find one that works. This process is crucial for understanding graphical representations of linear equations and inequalities, and it forms the basis for more complex optimization problems like linear programming, where you're looking for the best solution within a feasible region defined by multiple constraints.

Testing Point A: $(-3, -5)$

Alright, let's start with point A, which is $(-3, -5)$ . We need to plug these values ( $x = -3$ and $y = -5$ ) into both inequalities.

First inequality: $y > x - 3$ Substitute the values: $-5 > (-3) - 3$ Simplify: $-5 > -6$ Is this statement true? Yes, $-5$ is indeed greater than $-6$ . So, point A satisfies the first inequality.
Second inequality: $y \leq -2x + 5$ Substitute the values: $-5 \leq -2(-3) + 5$ Simplify: $-5 \leq 6 + 5$ Simplify further: $-5 \leq 11$ Is this statement true? Yes, $-5$ is less than or equal to $11$ . So, point A also satisfies the second inequality.

Since point A, $(-3, -5)$ , satisfies both inequalities, it is a solution to the system. We could stop here if this were a test and we were sure about our math! But, for the sake of thoroughness and learning, let's check the other points just to be absolutely sure and to reinforce the process.

It's super important to be careful with the substitution and the arithmetic, especially with negative numbers. A small slip-up can lead you to the wrong conclusion. For the first inequality, $-5 > -6$ is correct because on a number line, $-5$ is to the right of $-6$ . For the second, $-5 \leq 11$ is correct because $-5$ is much smaller than $11$ . This step confirms that point A is definitely in the solution set. This methodical approach is key. Don't guess, test! Each check builds your confidence and understanding of how these mathematical boundaries work together to define a solution space.

Testing Point B: $(3, -5)$

Next up is point B, $(3, -5)$ . Let's see if this guy makes the cut.

First inequality: $y > x - 3$ Substitute: $-5 > 3 - 3$ Simplify: $-5 > 0$ Is this true? Nope! $-5$ is definitely not greater than $0$ . This inequality fails.

Since point B fails the first inequality, we don't even need to check the second one. A solution must satisfy all inequalities in the system. Point B is not a solution.

This is a classic example of why testing both inequalities is crucial. Even if a point happened to satisfy one, it wouldn't be a valid solution for the system unless it satisfied all of them. Here, the y-value (-5) is much smaller than the x-value (3), and the first inequality requires y to be significantly larger than x. You can often get a feel for whether a point might work by a quick mental check, but the actual substitution is the only way to be certain. Don't let the negative numbers throw you off; just follow the order of operations and compare the results carefully. Failing here means it's outside the region defined by $y > x - 3$ , so it can't be in the overlapping solution set.

Testing Point C: $(5, -3)$

Let's put point C, $(5, -3)$ , to the test.

First inequality: $y > x - 3$ Substitute: $-3 > 5 - 3$ Simplify: $-3 > 2$ True or False? False. $-3$ is not greater than $2$ .

Point C fails the first inequality, so it's not a solution to the system. We can move on.

This is another instance where a point misses the mark. The inequality $y > x - 3$ essentially says the y-coordinate needs to be a certain amount larger than the x-coordinate. Here, $x=5$ and $y=-3$ . The condition requires $-3 > 5 - 3$ , which simplifies to $-3 > 2$ . This is clearly false. You can visualize this on a graph: the line $y = x - 3$ passes through points like $(3,0)$ and $(0,-3)$ . For $x=5$ , the line is at $y=2$ . The inequality $y > x - 3$ means we want points above this line. The point $(5, -3)$ is far below the point $(5, 2)$ , so it's definitely not in the desired region for the first inequality. This reinforces the idea that the relationships between x and y values are critical.

Testing Point D: $(5, 3)$

Finally, let's check out point D, $(5, 3)$ .

First inequality: $y > x - 3$ Substitute: $3 > 5 - 3$ Simplify: $3 > 2$ Is this true? Yes! $3$ is greater than $2$ . This inequality is satisfied.
Second inequality: $y \leq -2x + 5$ Substitute: $3 \leq -2(5) + 5$ Simplify: $3 \leq -10 + 5$ Simplify further: $3 \leq -5$ Is this true? Nope! $3$ is not less than or equal to $-5$ . This inequality fails.

Since point D fails the second inequality, it is not a solution to the system.

This last point, D, shows how a point can satisfy one inequality but not the other. For $x=5$ , the first inequality $y > x - 3$ requires $y > 2$ . Point D has $y=3$ , which works. However, the second inequality $y \leq -2x + 5$ requires $y \leq -2(5) + 5$ , meaning $y \leq -5$ . Point D has $y=3$ , which is much larger than $-5$ . So, $(5, 3)$ lies in the solution region of the first inequality but not the second. This confirms our earlier finding that A is the only point that worked for both.

Conclusion: The Winning Point

After carefully testing all the options, we found that only point A. $(-3, -5)$ satisfies both inequalities in the system:

$y > x - 3 -5 > -3 - 3 -5 > -6$ (True)
$y \leq -2x + 5 -5 \leq -2(-3) + 5 -5 \leq 6 + 5 -5 \leq 11$ (True)

So, the point that is a solution to the system is A. $(-3, -5)$ . Keep practicing these, and you'll be a pro at spotting solutions in no time! Math problems like these are all about methodical checking. Don't rush, and double-check your arithmetic, especially with negatives. Happy solving!