Problem 94 - Elementary Algebra

Question

Is the opposite of 7 smaller or larger than \(-6 ?\)

Step-by-Step Solution

Answer

The opposite of 7 is smaller than -6.

Step 1: Identify the Opposite

The opposite of a number is the number with the same magnitude but opposite sign. Since the number is 7, its opposite is \(-7\).

Step 2: Compare the Opposite with -6

Now that we know the opposite of 7 is \(-7\), we compare it with \(-6\). On the number line, \(-7\) is to the left of \(-6\), meaning it is smaller.

Key Concepts

Understanding Opposites

When we talk about opposites in math, we refer to numbers that are equidistant from zero on a number line but are on opposite sides. Think of opposites like mirror images with respect to zero. For any given number, its opposite is found by changing its sign.

• If you have a positive number, its opposite will be negative.
• Conversely, if you have a negative number, its opposite will be positive.

For instance, the opposite of 7 is (-7) , because you switch from positive to negative. This concept is crucial because understanding opposites helps in various operations and comparisons, especially when dealing with negative numbers.

Visualizing Numbers on a Number Line

A number line is a straightforward way to visually compare numbers, see their order, and understand relationships like opposites. When you place numbers on a number line:

• The numbers increase as you move to the right.
• The numbers decrease as you move to the left.

Zero acts as the central point between positive and negative numbers. The opposite numbers lie on either side of zero at equal distances. For example, 7 is located to the right of zero, while (-7) sits on the left side. Using the number line, it becomes easy to visually determine how numbers relate to each other, like seeing that (-7) is to the left of (-6) .

Comparing Numbers

Comparing numbers involves determining which number is greater or smaller. This is easier with a number line at hand. When two numbers are negative, the one closer to zero is larger. For example, with (-6) and (-7) :

• (-6) is closer to zero, meaning it is larger.
• (-7) is further from zero, making it smaller.

In general, for negative numbers:

• More negative means smaller.
• Less negative means larger.

Remembering these simple rules helps in performing various math tasks, like adding or subtracting negative numbers, and making accurate comparisons.