Problem 22 - College Algebra

Question

rewrite each expression without absolute value bars. $$ \frac{-7}{|-7|} $$

Step-by-Step Solution

Answer

-1

Step 1: Understanding Absolute Value

The absolute value of a number is always non-negative. In specific, the absolute value of -7 is 7. Given this understanding, the expression \(-7/|-7|\), can be rewritten as \(-7/7\).

Step 2: Calculating the simplified expression

We now have the expression -7/7. Both the numerator and denominator are the same number, but the numerator has a negative sign. As a result, -7 divided by 7 equals -1. The absolute value bars have been removed and the expression has been simplified.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step when solving many mathematical problems, as it makes them easier to understand and work with. When simplifying an expression like \(\frac{-7}{|-7|}\), the first step involves understanding what the absolute value bars represent. Absolute values convert any number into its non-negative counterpart. Thus, the expression inside the bars, \(-7\), becomes a 7.

After determining this, the expression simplifies from \(\frac{-7}{|-7|}\) to \(\frac{-7}{7}\). Notice that the absolute value bars have been replaced by a simple positive number. When simplifying expressions, always think of ways to break down complex symbols into familiar parts, like turning absolute values into their non-negative numbers.

Remember:

• Find absolute values first.
• Replace them in the expression.
• Simplify the expression to its most basic form.

This process not only simplifies mathematical expressions but also helps you focus on other aspects of the problem.

Dividing Integers

Dividing integers can seem tricky at first, but with a little practice, it becomes straightforward. When you have a division problem like \(\frac{-7}{7}\), you need to focus on both the numerator (the top number) and the denominator (the bottom number).

### Integer Division Rules

• Divide the absolute values of the numbers first: \(7 \div 7 = 1\).
• If the numbers were initially both positive or both negative, the result is positive.
• If one number is negative and the other is positive, the result is negative.

In the expression \(\frac{-7}{7}\), since the -7 is negative and the 7 is positive, the final answer is \(-1\). Following these guidelines makes dividing integers simple and efficient.

Negative Numbers

Negative numbers are numbers that are less than zero. They are an essential part of mathematics and commonly appear in various operations like adding, subtracting, multiplying, and dividing.

### Working with Negative Numbers

• Negatives are represented by a minus sign \((-\)).
• Adding a negative number is the same as subtracting its positive counterpart.
• Subtracting a negative number is like adding a positive number.
• Multiplying or dividing two negative numbers always yields a positive result.
• Multiplying or dividing a negative number with a positive number delivers a negative result.

Consider \(-7\): it shows the direction on the number line opposite to positive values such as 7. Being comfortable with negative numbers allows you to solve more complex problems confidently, especially when they appear in expressions with absolute values.