Finding Solutions: Which Point Works For The Inequality?

Hey math enthusiasts! Let's dive into the fascinating world of inequalities and figure out which point is a solution to the inequality y ≤ -2x + 3. Don't worry, it's not as scary as it sounds. We'll break it down step by step, making sure you grasp the concepts and feel confident in tackling these types of problems. Think of it as a fun puzzle where we're trying to find the missing piece that fits perfectly!

Understanding the Inequality: What Does It Mean?

Alright, before we jump into the solutions, let's make sure we're all on the same page about what y ≤ -2x + 3 actually represents. This is an inequality, which means it shows a relationship between two variables, x and y, but instead of saying they're equal, it says that y is either less than or equal to the expression -2x + 3.

So, what does this look like graphically? If we were to plot this inequality on a coordinate plane, we'd get a line (representing y = -2x + 3) and a shaded region. The line itself shows all the points where y equals -2x + 3. The shaded region indicates all the points where y is less than -2x + 3. Any point within the shaded region, or on the line, satisfies the inequality. Basically, the inequality defines a set of points (an area!) that meet a certain condition. The inequality is y ≤ -2x + 3. To fully understand this inequality, remember that the slope of the line is -2. That indicates how the y-value changes as we change x. The y-intercept is 3, which is the point where the line crosses the y-axis, when x equals 0. The '<=' sign means that the line itself is included as part of the solution.

Testing the Points: Let's Get Our Hands Dirty!

Now comes the fun part: testing the given points to see which one works. The key here is to substitute the x and y values of each point into the inequality and see if it holds true. If the inequality is true after the substitution, then that point is a solution. If not, it's not. Let's start plugging and chugging, shall we?

A. (0, 4)

Here, x = 0 and y = 4. Substituting these values into the inequality, we get:

4 ≤ -2(0) + 3 4 ≤ 0 + 3 4 ≤ 3

Is this true? Nope! 4 is not less than or equal to 3. So, the point (0, 4) is not a solution.

B. (1, 0)

Here, x = 1 and y = 0. Let's plug these values in:

0 ≤ -2(1) + 3 0 ≤ -2 + 3 0 ≤ 1

Is this true? Yes! 0 is less than or equal to 1. Therefore, the point (1, 0) is a solution.

C. (3, -2)

Here, x = 3 and y = -2. Let's substitute:

-2 ≤ -2(3) + 3 -2 ≤ -6 + 3 -2 ≤ -3

Is this true? No! -2 is not less than or equal to -3. Thus, the point (3, -2) is not a solution.

D. (5, -2)

Here, x = 5 and y = -2. Let's substitute:

-2 ≤ -2(5) + 3 -2 ≤ -10 + 3 -2 ≤ -7

Is this true? No! -2 is not less than or equal to -7. So, the point (5, -2) is not a solution.

The Correct Solution: Unveiling the Winner!

After testing each point, we found that only one point satisfied the inequality. The point (1, 0) is the solution to y ≤ -2x + 3. That means that when we substitute x = 1 and y = 0 into the inequality, it holds true. This point lies on the line defined by the equality y = -2x + 3. Any point above the line does not satisfy the inequality. Any point below the line satisfies the inequality. Therefore, the correct answer is option B. Congratulations if you got it right! If not, don't worry. This is an excellent opportunity to learn and improve. Remember that practice makes perfect, and with a little more practice, you'll be acing these inequality problems in no time. Keep up the excellent work!

Visualizing the Solution: Drawing It Out

To solidify our understanding, let's think about how this would look on a graph. Imagine plotting the line y = -2x + 3. This line has a slope of -2 (meaning it goes down as you move from left to right) and a y-intercept of 3 (it crosses the y-axis at the point (0, 3)). The inequality y ≤ -2x + 3 means we're looking for all the points that are either on the line or below it.

Think about it this way: if you were to draw a vertical line through the point (1, 0), it would intersect the y = -2x + 3 line at a point above (1, 0). All the points below that intersection on the vertical line would satisfy the inequality, including (1, 0). All the other points we tested (0, 4), (3, -2), and (5, -2) would fall outside of this shaded region, failing to meet the condition of the inequality.

This graphic representation provides a fantastic visual aid, allowing us to connect the algebraic concept of the inequality to its geometric counterpart. This is a very useful tool, offering a visual way to check if our solution makes sense. With this visual in mind, we can confirm that (1, 0) is, indeed, a solution. If you're into graphing, go ahead and plot the line and the points! It's a great way to confirm everything we've discussed and to further improve your comprehension of inequalities.

Tips and Tricks: Mastering Inequalities

Here are some helpful tips to keep in mind when working with inequalities:

• Always substitute the values carefully: Double-check that you're substituting the correct x and y values into the inequality. A small mistake can lead to a wrong answer.
• Understand the symbols: Remember what the inequality symbols mean:
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