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Chapter 2: Problem 68

Review operations with real numbers. Simplify. $$ \frac{1-(-7)}{-3-5} $$

Short Answer

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Step by step solution

01

Simplify the numerator

The numerator is the expression \(1 - (-7)\). Subtracting a negative is the same as adding the positive, so: \(1 - (-7) = 1 + 7 = 8\).
02

Simplify the denominator

The denominator is the expression \(-3 - 5\). Add the negative values together: \(-3 - 5 = -8\).
03

Divide the simplified numerator by the simplified denominator

Now divide the simplified numerator by the simplified denominator: \(\frac{8}{-8} = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions means making a mathematical expression as simple as possible. When you encounter a problem, break it into smaller parts and solve each one step by step. In the given example, \(\frac{1-(-7)}{-3-5}\), we first simplified the numerator and then the denominator before performing the division. This clear, step-by-step approach makes it much easier to handle complex expressions.
  • Identify individual parts of the expression that can be simplified.
  • Rearrange or combine like terms.
  • Simplify each component before putting them back together.

Taking these steps ensures you don't miss any details, and the final answer is accurate and easy to understand.
Numerator and Denominator
When dealing with fractions, understanding the numerator and denominator is essential. The numerator is the top part of the fraction, and the denominator is the bottom part.
For example, in the fraction \(\frac{8}{-8}\), 8 is the numerator, and -8 is the denominator. This helps indicate that we are dividing 8 by -8.
  • Always simplify the numerator and the denominator separately first.
  • Remember, subtracting a negative is the same as adding a positive.

In our exercise:
- The numerator is \ (1 - (-7)) \ which simplifies to 8.
- The denominator is \ (-3 - 5) \ which simplifies to -8.
Recognizing these components helps in clearly performing the operations required and avoiding common mistakes.
Dividing Integers
Dividing integers involves determining how many times one integer fits into another. The division can result in an integer, fraction, or decimal. In our problem, after simplifying the expression, we reached \(\frac{8}{-8}\).
Here鈥檚 how to handle such operations:
  • Divide the absolute values of the integers.
  • Determine the sign of the result. If the signs of the original integers are different, the result is negative. If they are the same, the result is positive.

Following this rule: \(\frac{8}{-8} = -1\).
  • The absolute value of 8 divided by 8 is 1.
  • Since the numerator and denominator have different signs, the result is negative.

Understanding these steps will allow you to confidently divide any integers you come across in your math problems.

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