Problem 59 - Prealgebra

Question

Give the opposite of each of the following numbers. $$-121$$

Step-by-Step Solution

Answer

The opposite of \(-121\) is \(121\).

Step 1: Understanding 'Opposite'

The opposite of a number is achieved by changing its sign. If the number is negative, its opposite will be positive, and vice versa. We're looking for a number that, when added to the original number, results in zero.

Step 2: Identifying the Number

We have the number \(-121\). It is a negative number.

Step 3: Changing the Sign

To find the opposite, we switch the sign of \(-121\) from negative to positive. This gives us \(121\).

Step 4: Verifying the Solution

Add the original number and its opposite: \(-121 + 121 = 0\). Since the result is zero, the opposite we found is correct.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, usually represented with a minus sign (-) before them. They are a fundamental part of the number system and are used to denote:

• Quantities that are below zero, such as temperatures below freezing.
• Debts in financial calculations or deficiencies in measurements.

In a number line representation, negative numbers are located to the left of zero. They are the counterparts to positive numbers, situated on the opposite side of zero. The more negative a number is, the further it is from zero. For instance, -5 is less than -2.

Negative numbers also play a vital role in various mathematical operations and equations. They are crucial in solving problems that require you to look into values that 'decrease' or are deducted from a total.

Exploring Positive Numbers

Positive numbers are greater than zero and are typically written without a positive sign (+) in front of them, as the presence of the sign is implied. They include:

• Whole numbers (1, 2, 3, ...).
• Fractions and decimals greater than zero (0.5, 2/3, etc.).

In math, positive numbers are considered more 'natural' as they often represent counts of objects, measures of distances, or quantities more than zero.

On a number line, you'd find positive numbers to the right of zero. This positioning symbolizes increases, gains, or amounts that add up. Positive numbers are pivotal in adding, subtracting, multiplying, and dividing equations, and understanding their role helps in efficiently solving different mathematical operations.

Techniques for Solving Equations

Solving equations involves finding the value of unknowns that make the equation true. Equations can include one or multiple operations such as addition, subtraction, multiplication, or division. Here are basic steps to solve equations, particularly around finding opposites:

• Identify the equation and separate the terms. Look for like terms or those which can be easily simplified.
• Use inverse operations to isolate the variable. For example, if the equation is in the form of \(x + a = b\), subtract \(a\) from both sides to get \(x = b - a\).
• Check your solution by substituting back into the original equation to ensure both sides are equal.

These techniques aid in solving linear equations as well as more complicated ones often used in algebraic expressions. By understanding each step, from identifying terms to applying inverse operations, you'll build confidence in solving for unknowns efficiently.