Inequalities: Find The Satisfying Point

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically tackling a super common challenge in algebra: figuring out which points actually satisfy a system of inequalities. This isn't just about crunching numbers; it's about understanding how these mathematical boundaries work and how to pinpoint the sweet spot where all conditions are met. So, grab your notebooks, maybe a snack, and let's break down this problem step-by-step. We've got a specific system of inequalities here:

$$y < 5x + 2$$

$$y \geq \frac{1}{2} x + 1$$

And we need to determine which of the given points – A. $(-1,3)$, B. $(0,2)$, C. $(1,2)$, or D. $(2,-1)$ – makes both of these inequalities true. It's like a mathematical treasure hunt, and only one point holds the key!

Understanding the Inequalities

Before we jump into testing the points, let's quickly chat about what these inequalities actually mean. Think of them as rules that define regions on a graph. The first inequality, $y < 5x + 2$, tells us that for any point $(x,y)$ to be valid, its y-coordinate must be strictly less than the value of $5x + 2$. This means the line $y = 5x + 2$ is a boundary, but the points on this line are not included. We'd represent this on a graph with a dashed line. The region satisfying this inequality is everything below that line.

The second inequality, $y \geq \frac{1}{2} x + 1$, is a bit different. Here, the y-coordinate must be greater than or equal to $\frac{1}{2} x + 1$. This means the line $y = \frac{1}{2} x + 1$ is also a boundary, but this time, the points on the line are included. We'd draw this boundary as a solid line. The region satisfying this inequality is everything on or above this line.

When we have a system of inequalities, like the one we're working with, we're looking for a point $(x,y)$ that simultaneously satisfies both conditions. Graphically, this would be the region where the shaded areas of both inequalities overlap. Our task today is to find which of the given coordinate pairs falls into this overlapping region.

Testing the Points: The Core of the Solution

Alright, guys, this is where the real action happens. To find out which point satisfies the system, we need to plug the x and y values of each option into both inequalities. If a point makes both inequalities true, then bingo! That's our answer. If it fails even one, it's out of the running.

Let's start with Option A: $(-1,3)$.

• First Inequality: $y < 5x + 2$ Substitute $x = -1$ and $y = 3$: Is $3 < 5(-1) + 2$? $3 < -5 + 2$ $3 < -3$. This is false.

Since the first inequality is false for point A, we don't even need to check the second one. Point A does not satisfy the system. Onwards to the next!

Option B: $(0,2)$.

• First Inequality: $y < 5x + 2$ Substitute $x = 0$ and $y = 2$: Is $2 < 5(0) + 2$? $2 < 0 + 2$ $2 < 2$. This is also false. Remember, the inequality is strictly less than ($<$), not less than or equal to ($\leq$). So, points on the line $y=5x+2$ are not included.

Another one bites the dust! Point B doesn't make the cut either. Keep pushing, we'll find it!

Option C: $(1,2)$.

• First Inequality: $y < 5x + 2$ Substitute $x = 1$ and $y = 2$: Is $2 < 5(1) + 2$? $2 < 5 + 2$ $2 < 7$. This is true! Awesome, point C is still in the game.

• Second Inequality: $y \geq \frac{1}{2} x + 1$ Substitute $x = 1$ and $y = 2$: Is $2 \geq \frac{1}{2}(1) + 1$? $2 \geq \frac{1}{2} + 1$ $2 \geq 1.5$. This is also true!

Hold up! Point C satisfies both inequalities. It looks like we might have found our winner. But, for thoroughness and to really nail this, let's just quickly check the last option.

Option D: $(2,-1)$.

• First Inequality: $y < 5x + 2$ Substitute $x = 2$ and $y = -1$: Is $-1 < 5(2) + 2$? $-1 < 10 + 2$ $-1 < 12$. This is true. Okay, point D passes the first test.

• Second Inequality: $y \geq \frac{1}{2} x + 1$ Substitute $x = 2$ and $y = -1$: Is $-1 \geq \frac{1}{2}(2) + 1$? $-1 \geq 1 + 1$ $-1 \geq 2$. This is false.

So, point D fails the second inequality. That confirms our suspicion: Point C $(1,2)$ is the one! It's the only point that successfully navigates the boundaries set by both inequalities.

Why This Matters: Visualizing the Solution

Understanding how to test these points is crucial, but it's even cooler when you can visualize it. Imagine graphing these two lines. The first line, $y = 5x + 2$, has a steep positive slope and a y-intercept at $(0,2)$. The inequality $y < 5x + 2$ shades the entire region below this dashed line. The second line, $y = \frac{1}{2} x + 1$, has a gentler positive slope and a y-intercept at $(0,1)$. The inequality $y \geq \frac{1}{2} x + 1$ shades the region on and above this solid line.

The solution to the system is the area where these two shaded regions overlap. Our point $(1,2)$ lies within this specific overlap zone. When you plot these lines and shade the regions, you'd see that $(1,2)$ is in the