Problem 10 - Algebra 1: Concepts and Skills

Question

Find the opposite of the number. $$ \left|-\frac{1}{5}\right| $$

Step-by-Step Solution

Answer

The opposite of the absolute value of -1/5 is 1/5.

Step 1: Understand the absolute value

An absolute value, indicated by the two vertical lines \( | | \), is essentially a function that takes the 'distance' a number is away from zero on a number line, disregarding direction. Thus, it makes any negative number positive and leaves any positive number as it is.

Step 2: Apply the absolute value function on the given number

Here, we are given a number inside this function as -1/5. Thus when we apply the absolute value function, it turns -1/5 into 1/5.

Step 3: Finalize the solution

Therefore, the opposite of the absolute value of -1/5, is simply the result itself, which is 1/5.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are always written with a minus (-) sign in front of them, such as -1, -2, or -1/5. Negative numbers are used in various contexts, like temperatures below freezing or debts below zero dollars.
They are an essential part of mathematics and show up whenever we need to express values less than zero. On a number line, negative numbers are located to the left of zero. This directionality signifies that they are less than positive numbers. For example, -1 is less than 0, and -2 is even less than -1.

• The more left a number is on the number line, the smaller it is.
• Negative numbers can be added, subtracted, multiplied, and divided just like positive numbers.
• When adding a negative number, it's the same as moving left on the number line.

Exploring Opposite Numbers

In mathematics, opposite numbers are two numbers that have the same numerical value but different signs. For example, the opposite of 5 is -5, and the opposite of -1/5 is 1/5. Opposite numbers are equidistant from zero on a number line.
When considering opposites, think about them as mirror images of each other. If one number moves to the right from zero, its opposite moves an equal distance to the left.

• To find the opposite of a number, simply change its sign.
• The opposite of zero is still zero, as it is the midpoint on the number line.
• Opposite numbers always sum up to zero. For example, 5 + (-5) = 0.

Reading a Number Line

A number line is a visual representation used in math to show numbers in a straight line. It includes all integers, positive and negative, and allows us to easily understand relationships between them.
On the number line, numbers increase as you move to the right and decrease as you move to the left. Zero is typically placed at the center.

• Every move to the right represents a positive increase.
• Each move to the left indicates a decrease or a move to a smaller number.
• Number lines help us see how absolute value works: regardless of direction, the absolute value measures distance from zero.

Understanding how to read and use a number line can help visualize absolute value problems, show distance between numbers, and simplify addition and subtraction of integers.