Question

If -3 and 5 are the coordinates of two points on the real number line, the distance between these points is ____

Short answer

The distance is 8.

Step-by-step solution

- Understand the problem

We are given two points on the real number line: -3 and 5. We need to find the distance between these two points.

- Recall the formula for distance on a number line

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

- Substitute the given points into the formula

Substitute \(a = -3\) and \(b = 5\) into the formula: \[ \text{Distance} = |-3 - 5| \]

- Simplify the expression

Simplify the expression inside the absolute value: \[ -3 - 5 = -8 \] Then, take the absolute value of -8, which is 8. \[ | -8 | = 8 \]

- State the final answer

The distance between the two points -3 and 5 is 8.

Key concepts

absolute value

When working with distances on the number line, understanding the absolute value is essential. Absolute value is a measurement of the magnitude of a number, regardless of its direction on the number line. It is denoted by vertical bars: \(| x |\). The absolute value of a number is its distance from zero. For example, the absolute value of -8 is 8 because -8 is 8 units away from zero.

Here are some quick points about absolute value:

• Absolute value is always non-negative.
• It turns negative values into positive ones.
• Zero remains zero because its distance from itself is zero.

To find the distance between two numbers \(a\) and \(b\) on a number line, you can use the formula: \[ \text{Distance} = |a - b| \] This formula helps determine how far apart two points are without considering their direction.

subtracting integers

Subtracting integers can be tricky, but it's a crucial part of finding distances on the real number line. Subtraction involves finding the difference between two numbers. Here's a simple example: when you subtract a smaller number from a larger one, the result is positive. For instance, \5 - 3 = 2\.

But, what if you subtract a larger number from a smaller one? The result is negative, such as \3 - 5 = -2\.

In our problem, when subtracting the points \-3\ and \5\, you set it up as: \(-3 - 5\). This becomes more manageable if you remember that subtracting a positive number is like adding its negative: \(-3 - 5 = -3 + (-5) = -8\).

Quick points about subtracting integers:

• Subtracting a positive is the same as adding a negative.
• Subtracting a negative is the same as adding a positive.
• Always pay attention to the signs of the numbers involved.

real number line

The real number line is a visual representation of all real numbers. Numbers increase as you move to the right and decrease as you move to the left. Points on the line correspond to numbers. For example, -3 is to the left of 0, and 5 is to the right of 0.

When finding distances between points on the real number line, it's helpful to visualize or draw the line. This makes it easier to understand how far apart the numbers are.

Key tips for understanding the real number line:

• It includes all integers, fractions, and irrational numbers.
• Distances on the line are always nonnegative.
• Distances can be calculated by counting units or using the absolute value formula.

Visualizing the real number line helps simplify solving problems. It can make abstract concepts more concrete. Always remember, the number line is a straight path where every point has a unique coordinate corresponding to a real number.