## Explain Why |−3| + |9| Represents the Distance Between the Points (−3, −5) and (9, −5).

When it comes to determining the distance between two points in a coordinate plane, the formula that often comes to mind is the Pythagorean theorem. However, in the case of finding the distance between the points (-3, -5) and (9, -5), we can use a different approach using absolute value. Let me explain why |−3| + |9| represents this distance.

First, let’s consider the x-coordinates of the two points: -3 and 9. Taking their absolute values gives us 3 and 9 respectively. Since both points lie on the same horizontal line (y = -5), we don’t need to account for any vertical movement.

Now, by adding these absolute values together (3 + 9 = 12), we obtain a result of 12. This indicates that there is a total horizontal displacement of 12 units between the two given points.

Hence, we can conclude that |−3| + |9| represents the distance between (-3, -5) and (9, -5). In this case, it signifies a horizontal distance of 12 units without considering any vertical change.

## Definition of Absolute Value

Distance Formula in a Coordinate Plane

Let’s delve into the fascinating concept of the distance formula in a coordinate plane and explore why |−3| + |9| represents the distance between the points (−3, −5) and (9, −5). The distance formula is a fundamental tool that enables us to calculate the distance between two points on a coordinate plane.

To understand this formula better, let’s break it down step by step:

- Identify the coordinates: In our case, we have two points: Point A with coordinates (-3, -5) and Point B with coordinates (9, -5).
- Determine the horizontal difference: We subtract the x-coordinate of Point A from the x-coordinate of Point B to find the horizontal difference. In this case, 9 – (-3) = 12.
- Determine the vertical difference: Similarly, we subtract the y-coordinate of Point A from the y-coordinate of Point B to find the vertical difference. In this case, -5 – (-5) = 0.
- Apply absolute value: To ensure that our result is always positive, we take the absolute value of both differences obtained in steps 2 and 3.

Now let’s put it all together! The distance between two points can be calculated using this formula:

Distance = |horizontal difference| + |vertical difference|

Applying this to our example:

Distance = |12| + |0|

Since any number plus zero equals itself and taking an absolute value does not change positive numbers:

Distance = 12

Hence, |−3| + |9| represents a distance of 12 units between point A (-3,-5) and point B (9,-5). This demonstrates how we can use absolute values to compute distances accurately on a coordinate plane.

## Explanation of the Given Points

To understand why |−3| + |9| represents the distance between the points (−3, −5) and (9, −5), let’s break it down step by step.

First, we need to find the absolute value of -3. The absolute value of a number is its distance from zero on a number line. In this case, |-3| equals 3.

Next, we find the absolute value of 9, which is simply 9 since it’s already positive.

Now that we have both absolute values – 3 and 9 – we add them together: 3 + 9 = 12.

So, why does this sum represent the distance between the given points? Well, in mathematics, when we’re dealing with coordinates on a plane (like (-3,-5) and (9,-5)), we use what’s called the distance formula to calculate their distance.

The distance formula states that for two points (x1,y1) and (x2,y2), the distance between them is given by √((x2-x1)^2 + (y2-y1)^2).

In our case, both y-coordinates are -5. Therefore, we can simplify the formula to √((x2-x1)^2). Plugging in x1 = -3 and x2 = 9 gives us √((9-(-3))^2), which simplifies further to √(12^2).

And here’s where our earlier calculation comes into play! Recall that |−3| + |9| equals 12. So our final expression becomes √(12^2) or simply sqrt(144).

The square root of 144 is exactly equal to its positive value: 12. Hence, |−3| + |9| indeed represents the distance between (-3,-5) and (9,-5), which is 12 units.

This connection between the absolute values and the distance formula allows us to find distances efficiently, without having to deal with negative numbers. It’s a useful technique in many mathematical and real-world applications.