Problem 52 - Elementary Algebra

Question

On the number line, how many units between -3 and \(2 ?\)

Step-by-Step Solution

Answer

Answer: 5 units

Step 1: Identify the given points

The points given on the number line are -3 and 2.

Step 2: Subtract the smaller number from the larger number

To find the distance between -3 and 2, subtract the smaller number (-3) from the larger number (2): 2 - (-3)

Step 3: Simplify the expression

Simplify the expression by changing the subtraction of a negative number to addition: 2 + 3

Step 4: Calculate the result

Add the two numbers: 2 + 3 = 5

Step 5: Write the final answer

There are 5 units between -3 and 2 on the number line.

Key Concepts

Number Line

The number line is a fundamental tool in mathematics that visually represents numbers as points on a straight line. It's a simple yet powerful concept that can help you understand mathematical operations and the relationships between numbers. Positive numbers are typically placed to the right of the origin, marked as zero, while negative numbers extend to the left. The number line is not just a representation of whole numbers; it also includes fractions and decimals at precise intervals between whole numbers.

Understanding how to navigate the number line is crucial for many areas of mathematics, including calculating distance. When calculating the number of units between two points, such as -3 and 2, you essentially measure the length of the segment that connects these points on the number line. This distance is always positive, as it represents a magnitude without direction.

Negative Numbers

Negative numbers are denoted by a minus sign (-) and represent quantities less than zero. They are used to describe values that are lower in magnitude relative to a reference point or to express a loss or absence of a quantity. On the number line, negative numbers lie to the left of zero.

One of the critical features of negative numbers is that they are an extension of the familiar positive numbers, allowing us to express and solve problems involving directions or values that move below a starting point. When performing subtraction on the number line, moving to the left signifies subtracting, and moving to the right signifies adding. Understanding negative numbers is important for grasping the concept of subtraction of integers.

Subtraction of Integers

Subtraction of integers might seem tricky at first, especially when dealing with negative numbers. To subtract an integer, think of it as moving backwards on the number line from a starting point. When we subtract a positive number, we move left, and when we subtract a negative number, it's as if we're moving right.

For example, if we take the expression 2 - (-3), we're effectively moving to the right from 2 for three units on the number line because subtracting a negative is the same as adding its positive counterpart. Therefore, when subtracting integers, remember that two negatives make a positive. This is why the distance calculation in the given exercise changes the subtraction of a negative number to the addition of a positive number.

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction, and is always non-negative. It's indicated by two vertical bars surrounding the number; for example, |x| is the absolute value of x. The absolute value measures how far a number is from zero, so |3| and |-3| both equal 3.

In context with finding the distance on the number line, absolute value is extremely useful. For the exercise about the distance between -3 and 2, we can use absolute value to understand that the distance is five units since |2 - (-3)| equals |2 + 3|, which simplifies to 5. Whenever you're tasked with finding the distance between two points on the number line, absolute value will ensure that you get a positive value, signifying the actual distance.