Problem 26 - College Algebra

Question

Distance of 10 units from the number 4

Step-by-Step Solution

Answer

The numbers 10 units from 4 are 14 and -6.

Step 1: Understand Distance on a Number Line

Distance on a number line is the positive difference between two points. It can be in either direction from a starting point. Here, we need to find points that are 10 units away from the number 4.

Step 2: Determine Positive Direction Distance

To find the position 10 units to the right of 4, we add 10 to 4. This gives us the point: \[ 4 + 10 = 14 \]. So, one number that is 10 units away from 4 is 14.

Step 3: Determine Negative Direction Distance

To find the position 10 units to the left of 4, we subtract 10 from 4. This gives us the point: \[ 4 - 10 = -6 \]. So, another number that is 10 units away from 4 is -6.

Step 4: Conclusion

There are two numbers which are at a distance of 10 units from the number 4 on the number line. These numbers are 14 and -6.

Key Concepts

Positive Difference

Distance on a number line is always measured as a positive difference. This means that when we calculate how far apart two numbers are on a number line, we always end up with a positive number. Positive difference ensures that the distance is never represented as negative.
To find the positive difference between two numbers, you simply:

• Subtract the smaller number from the larger number.
• If you're unsure which is larger, use the absolute value of the difference.

For example, if comparing points 14 and 4, you would calculate:$\backslash [\backslash text\{Absolute\; Distance\}\; =\; |14\; -\; 4|\; =\; 10\backslash ]$This positive difference confirms how far apart the numbers are on the number line.

Number Line

A number line is a straight horizontal line that represents numbers at equal intervals or distances. It's a handy tool for visualizing distance, positions, and the relationships between numbers.

• Numbers to the right are greater, while numbers to the left are smaller.
• The distance between any two points can be easily measured using positive difference.
• The number line extends infinitely in both directions.

For our current exercise, our focus is on finding two points that are 10 units away from the number 4 on this number line. This visual can help you clearly observe both positions at 14 and -6.

Units from a Point

When you say "units from a point," it refers to the absolute distance from a specified number, moving in either direction. It's like asking "How far is something from here?"
For example, when given a point on a number line, like 4, and asked to find points 10 units away:

• To move right, you "add" units to the starting point. $\backslash [4\; +\; 10\; =\; 14\backslash ]$
• To move left, you "subtract" units from the starting point. $\backslash [4\; -\; 10\; =\; -6\backslash ]$

This concept helps in determining and interpreting where points lie with reference to a given number.

Algebraic Distance

Algebraic distance is used to determine positions in an equation form, showing how far apart numbers are on a number line. You represent these changes in position algebraically by adding or subtracting values. Given the number 4 as your starting point:

• The algebraic expression for moving 10 units right is \(4 + 10 = 14\)
• The algebraic expression for moving 10 units left is \(4 - 10 = -6\)

Algebraic distance helps in solving problems systematically using equations, making it easier to find various points at a set distance from any given number.