Problem 109 - Elementary and Intermediate Algebra

Question

What two numbers are a distance of 6 away from \(-3\) on the number line?

Step-by-Step Solution

Answer

The numbers are 3 and -9.

Step 1: Understanding the Problem

We are tasked with finding two numbers that are 6 units away from -3 on the number line. This implies that starting from -3, we either move 6 units to the right or 6 units to the left to find the two numbers.

Step 2: Calculating the First Number

To find the first number, we move 6 units to the right of -3. Moving right on the number line means we are adding 6 to -3. Thus, \[-3 + 6 = 3\].

Step 3: Calculating the Second Number

To find the second number, we move 6 units to the left of -3. Moving left on the number line means we are subtracting 6 from -3. Thus, \[-3 - 6 = -9\].

Step 4: Verifying the Distance

Verify that both numbers are actually 6 units away from -3. For 3, the distance is \(|3 - (-3)| = |3 + 3| = 6\). For -9, the distance is \(|-9 - (-3)| = |-9 + 3| = |-6| = 6\). This confirms that both numbers are correct.

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics. It is the measure of how far a number is from zero on the number line. The key to understanding absolute value is to remember that it is always a non-negative number.

When solving problems, like the distance between two points on a number line, absolute value becomes particularly useful. To calculate the distance between two numbers, take the absolute value of their difference. For example, if you want to find how far 3 is from \(-3\), compute \(|3 - (-3)| = |3 + 3| = 6\). This gives the total distance of 6 units.

Absolute value can be demonstrated with simple rules:

• The absolute value of a positive number is the number itself: \(|5| = 5\).
• The absolute value of a negative number is its positive counterpart: \(|-5| = 5\).
• The absolute value of zero is zero: \(|0| = 0\).

Understanding absolute value helps clarify why distances are always positive and reinforces its utility in determining distances on the number line.

Adding and Subtracting Integers

Adding and subtracting integers can sometimes be confusing, but with practice, it becomes intuitive. When dealing with integer operations, the number line is a powerful tool. It helps visualize moving left and right depending on whether you are adding or subtracting.

Consider these guidelines:

• Adding a positive integer: Move to the right on the number line.
• Subtracting a positive integer: Move to the left.
• Adding a negative integer: This is the same as subtracting the absolute value of that integer.
• Subtracting a negative integer: This is like adding the absolute value of that integer.

For example, when we add 6 to \(-3\), we move 6 spaces to the right, leading us to 3. When we subtract 6 from \(-3\), we move 6 spaces to the left, reaching \(-9\). Practice these operations with different numbers to get comfortable with the concepts.

Number Line Concept

The number line is a visual representation that helps illustrate the size and order of numbers. It's a straight horizontal line where every point corresponds to a number. Numbers increase as you move to the right and decrease as you go to the left.

A few points to remember when using a number line:

• Zero is typically at the center, separating positive numbers on the right from negative numbers on the left.
• The distance between each consecutive number is uniform.
• Useful for illustrating addition and subtraction visually.
• Aids in understanding the notion of absolute value and distances.

In the exercise that involves finding numbers at a specific distance from \(-3\), the number line helps clearly show the positions of potential solutions. By counting units to the left or right, we can quickly see how numbers are spaced and confirm any computed results with absolute accuracy.