Problem 74 - College Algebra Essentials

Question

Express the distance between the given numbers using absolute value. Then fi nd the distance by evaluating the absolute value expression. -5.4 and -1.2.

Step-by-Step Solution

Answer

The distance between -5.4 and -1.2 is 4.2.

Step 1: Understand absolute value

The absolute value of a number is its distance from 0 on the number line. It is denoted by | |. By definition, if a number is positive, its absolute value is the number itself. If the number is negative, its absolute value is the number without the negative sign.

Step 2: Determine the expression for distance

The distance between any two numbers on a number line is the absolute value of their difference. So, the distance between -5.4 and -1.2 can be expressed as |(-5.4) - (-1.2)| or |-5.4 - (-1.2)|. It's important to notice that the minus sign before 1.2 turns into a plus when distributed.

Step 3: Evaluate the absolute value expression

Simplify the expression inside the absolute value brackets before applying the absolute value. |-5.4 - (-1.2)| = |-5.4 + 1.2| = |-4.2|. The absolute value of -4.2 is 4.2.

Key Concepts

Distance Between Numbers

To understand the distance between numbers on a number line, it is crucial to remember the concept of absolute value. Distance in mathematics is always a non-negative quantity. When we talk about the distance between two numbers, such as -5.4 and -1.2, we refer to how far apart they are on the number line without considering direction.

The distance between two numbers can be found by taking their difference and then applying the absolute value function. This is because absolute value measures how far a number is from zero, ignoring whether that distance is forwards or backwards on the number line. Therefore, the equation to finding the distance between -5.4 and -1.2 is expressed as \(|-5.4 - (-1.2)|\).

This form ensures we calculate the true distance between these points while acknowledging that subtracting a negative number is the same as adding its positive value.

Number Line

A number line is a visual representation of numbers progressing either to the left or right of zero. It's an excellent tool for understanding the concept of absolute value and calculating distance. On a number line, distances are always calculated in terms of movement units, disregarding direction.

For example, our task is to understand the distance between -5.4 and -1.2 on this line. By placing -5.4 and -1.2 on the number line, we can see that -5.4 lies to the left of -1.2. The numerical difference between these two points, without regard to direction, will show us how far they are apart. Applying absolute value to this calculation gives us a clear, positive expression of that distance.

• Visualize -5.4 and -1.2 as separate points.
• Calculate the steps needed to move from -5.4 to -1.2.
• Use absolute value to express this calculated distance positively.

Evaluating Expressions

Evaluating expressions involves simplifying mathematical statements to find their numerical values. When dealing with absolute value expressions, the goal is to break down the steps systematically until you can find a non-negative result.

In the case of finding the distance between -5.4 and -1.2, we start by setting up the expression \(|-5.4 - (-1.2)|\). We evaluate this by simplifying inside the absolute value first:

1. Recognize that \(-(-1.2)\) turns into \(+1.2\), yielding the expression \(-5.4 + 1.2\).
2. Calculate \(-5.4 + 1.2\) to get \(-4.2\).
3. Finally, apply the absolute value, turning \(-4.2\) into \(4.2\).

This process ensures that regardless of the original direction or 'sign' of the difference, the end result is a distance, a universally positive measure.