Problem 83 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ 315 \text { and }-213 $$

Step-by-Step Solution

Answer

The distance between 315 and -213 is 528.

Step 1: Understand the Problem

We need to find the distance between two numbers on a number line. The numbers are 315 and -213.

Step 2: Identify Positions on Number Line

Visualize the number line with -213 located to the left of 315. Both positions can be imagined as points on this line.

Step 3: Use Distance Formula

The distance between two points on a number line is calculated using the formula: \[\text{Distance} = |a - b|\] where \(a\) and \(b\) are the two numbers.

Step 4: Substitute Values

Substitute \(a = 315\) and \(b = -213\) into the distance formula: \[\text{Distance} = |315 - (-213)|\]

Step 5: Simplify the Expression

Simplify inside the absolute value brackets: \[315 - (-213) = 315 + 213\]This results in:\[|528|\]

Step 6: Calculate Absolute Value

The absolute value \(|528|\) is simply 528, since absolute value represents the distance which cannot be negative.

Step 7: Conclusion

Therefore, the distance between 315 and -213 on a number line is 528.

Key Concepts

Absolute Value

Absolute value is a measure of how far a number is from zero, regardless of direction on the number line. It compels us to focus on the magnitude, not the sign of a number. Imagine the absolute value as stripping away any negatives, leaving only the distance.

• It is denoted by two vertical bars: \(|a|\), where \(a\) is any number.
• This value always results in a non-negative number.

For example, the absolute value of -5 and 5 is the same, \(5\), since both are 5 units away from zero on a number line. Harnessing this concept helps us confidently solve distance problems, as distances themselves can’t be negative.

Positioning on Number Line

Locating numbers like 315 and -213 on a number line helps visualize their relative distances. The number line is a straight line where numbers increase to the right and decrease to the left from zero.

• Negative numbers, like \(-213\), appear on the left of zero.
• Positive numbers, like \(315\), appear on the right of zero.

By understanding how these numbers are positioned, one can intuitively sense that larger positive numbers will always be farther right compared to smaller or negative numbers. This mental image aids in accurately applying the distance formula.

Distance Formula

The distance formula is a straightforward but powerful tool for finding the distance between two points on a number line. The elegance of the formula \[\text{Distance} = |a - b|\] lies in its simplicity:

• By taking the difference \(a - b\), we identify how far two numbers are from each other in either direction.
• The absolute value \(| ext{difference}|\) ensures this distance is non-negative, reflecting true physical distance.

For example, to find the distance between 315 and -213, substituting these values into the formula results in \[\text{Distance} = |315 - (-213)|\]. Simplifying it step-by-step reveals a distance of 528. This formula applies universally across all number line distance problems, cementing its usefulness.