Question

On the number line, what is the distance between -1.45 and -8.34 ? A. -9.79 B. -6.89 C. 6.89 D. 9.79

Short answer

C. 6.89

Step-by-step solution

Identify the points on the number line

The two points given in the problem are -1.45 and -8.34. Locate these points on the number line.

Understand the formula for distance between two points

The distance between two points on a number line is given by the absolute value of the difference between the two points. Formula: \( \text{Distance} = |a - b| \).

Plug in the values

Using the points from Step 1, we get \(a = -1.45\) and \(b = -8.34\). So, \( \text{Distance} = |-1.45 - (-8.34)| \).

Simplify the expression

Calculate the difference inside the absolute value: \( -1.45 - (-8.34) = -1.45 + 8.34 = 6.89 \).

Apply the absolute value

The absolute value of 6.89 is \( |6.89| = 6.89 \).

Choose the correct answer

The distance between -1.45 and -8.34 is 6.89. The correct answer is C. 6.89.

Key concepts

absolute value

The concept of absolute value is essential when dealing with distances on a number line. Absolute value refers to how far a number is from zero, regardless of direction. Think of it as the 'distance' of a number from zero.

For example, the absolute value of both 3 and -3 is 3, written as \( |3| = 3 \) and \( |-3| = 3 \). The principle of absolute value ensures that distance is always a positive quantity.

This becomes critical in calculations because distance should not be a negative value. When we find the distance between two points, we first find their difference and then take the absolute value.

In the example above, the distance between -1.45 and -8.34 was calculated by finding the difference first. The difference was 6.89. Even if the value inside the absolute value was negative, applying the absolute value rule guarantees a positive distance.

difference of numbers

To understand the distance between two points, we need to know the difference between those numbers. The difference is simply obtained by subtracting one number from another.

For two points \(a\) and \(b\) on the number line, the difference is found using the formula \(a - b\). This formula helps us determine how far one point is from the other.

In our case, the points were -1.45 and -8.34. The difference was calculated as follows: \( -1.45 - (-8.34) = -1.45 + 8.34\). Notice how subtracting a negative is the same as adding a positive. This step gives us the difference of 6.89.

It’s crucial to calculate this carefully to use in further steps.

number line

A number line is a visual representation used to understand the relative position of numbers. It has zero at the center, with positive numbers extending to the right and negative numbers to the left.

When working with distances on the number line, the absolute value and the difference between numbers become clearer. Let's take our example: the points -1.45 and -8.34. Placing these on the number line, -1.45 is further to the right than -8.34.

To find the distance between them, we’re essentially counting how many units we move on the number line from -1.45 to -8.34 or vice versa. This is why we use the absolute value of their difference.

By visualizing these points, the idea of distance as the 'total space' between numbers, regardless of direction, becomes easier to grasp.