Inequality Graphs: Finding The False Statement

Oct 22, 2025 by Dimemap Team 47 views

Let's dive into inequality graphs! This topic often appears in math, and understanding how to interpret them is super useful. We'll tackle an inequality problem step-by-step, focusing on how to identify the incorrect statement about its graph. So, let's get started and make things crystal clear!

Understanding the Inequality

Okay, guys, first things first: the inequality we're dealing with is $3x - 6y > 12$ . To really get a grip on this, we need to simplify it and see what it tells us about the relationship between x and y. Let's divide the entire inequality by 3 to make the numbers smaller and easier to handle. This gives us: $x - 2y > 4$ . Now, this looks a bit more manageable, right?

But wait, there's more! To visualize this inequality as a graph, we often want to express it in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. So, let's rearrange our simplified inequality to solve for y. Starting with $x - 2y > 4$ , we can subtract x from both sides to get $-2y > -x + 4$ . And now, to isolate y, we'll divide both sides by -2. Remember that dividing by a negative number flips the inequality sign! So we get: $y < (1/2)x - 2$ .

Now, what does this tell us? The inequality $y < (1/2)x - 2$ represents all the points (x, y) on the coordinate plane that lie below the line y = (1/2)x - 2. The line itself is not included because we have a 'less than' sign (<) rather than a 'less than or equal to' sign (≤). So, when we graph this, we'll draw a dashed line to indicate that the points on the line are not part of the solution. And everything below that dashed line is the solution set. Got it? Great! Let's move on to analyzing the statements about this graph.

Analyzing the Statements

Now that we understand the inequality $y < (1/2)x - 2$ , we can evaluate the given statements to see which one is incorrect. Here are the statements:

A. The x-intercept is $(4, 0)$ . B. The y-intercept is $(0, -2)$ . C. The point $(0, 0)$ is not a solution.

Let's check each statement one by one.

Statement A: The x-intercept is (4, 0)

The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always 0. To find the x-intercept, we set y = 0 in the equation of the line x - 2y = 4 (we use the equation of the line, not the inequality, to find the intercepts). So, we have x - 2(0) = 4, which simplifies to x = 4. Therefore, the x-intercept is indeed (4, 0). So, statement A is correct.

Statement B: The y-intercept is (0, -2)

The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is always 0. To find the y-intercept, we set x = 0 in the equation of the line x - 2y = 4. So, we have 0 - 2y = 4, which simplifies to -2y = 4. Dividing both sides by -2, we get y = -2. Therefore, the y-intercept is indeed (0, -2). So, statement B is also correct.

Statement C: The point (0, 0) is not a solution

To check if the point (0, 0) is a solution to the inequality $y < (1/2)x - 2$ , we substitute x = 0 and y = 0 into the inequality. This gives us 0 < (1/2)(0) - 2, which simplifies to 0 < -2. Is this true? Nope! 0 is definitely not less than -2. This means that the point (0, 0) does not satisfy the inequality, and therefore it is not a solution. So, statement C is also correct.

Wait a minute! All the statements are correct? That's unexpected. Let's go back to the original inequality and double-check our work. The original inequality is $3x - 6y > 12$ . We divided by 3 to get $x - 2y > 4$ , and then rearranged to get $y < (1/2)x - 2$ . Everything seems correct so far. The x-intercept is (4,0) because 3(4) - 6(0) > 12 is true. The y-intercept is (0, -2) because 3(0) - 6(-2) > 12 is true. And (0, 0) is not a solution because 3(0) - 6(0) > 12 is false.

Okay, I think I found the problem. The problem statement says: Pernyataan berikut yang tidak benar mengenai grafik pertidaksamaan tersebut adalah

Which translates into: Which of the following statements about the graph of the inequality is NOT true?

Since all statements are true, this means there might be an error in the original question. Let's assume, for the sake of argument, that the original question was intending to ask: Which of the following statements is FALSE about the inequality $3x - 6y > 12$ ?

Let's analyze again:

A. The x-intercept is $(4, 0)$ . Plug in (4,0) into the inequality: $3(4) - 6(0) > 12 -> 12 > 12$ . This is FALSE. Because $12 = 12$ , not greater than. B. The y-intercept is $(0, -2)$ . Plug in (0, -2) into the inequality: $3(0) - 6(-2) > 12 -> 12 > 12$ . This is FALSE. Because $12 = 12$ , not greater than. C. The point $(0, 0)$ is not a solution. Plug in (0,0) into the inequality: $3(0) - 6(0) > 12 -> 0 > 12$ . This is TRUE, (0,0) is not a solution.

Therefore, the answer is A and B. Either there is a typo in the question, or the question is trying to trick you! The key here is to pay close attention to the wording of the problem. Are we looking for something that is TRUE or FALSE?

Conclusion

So, to wrap things up, understanding inequality graphs involves simplifying the inequality, visualizing it as a line on a graph, and then testing different points to see if they satisfy the inequality. It's about being meticulous and double-checking your work, especially when dealing with tricky questions! Remember to always read the question carefully and determine exactly what is being asked. Keep practicing, and you'll become a pro at this in no time!