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Chapter 1: Problem 11

Evaluate each expression. $$\frac{-4+(7-9)}{(-9-7)+4}$$

Short Answer

Expert verified
The expression simplifies to \(0.5\).

Step by step solution

01

Solve inside parentheses

First perform the operations inside the parentheses. In the numerator, calculate \(7 - 9\) to get \(-2\). In the denominator, calculate \(-9 - 7\) to get \(-16\). So the expression becomes \(\frac{-4 + (-2)}{-16 + 4}\).
02

Perform addition

Then, calculate \(-4 + (-2)\) in the numerator to get \(-6\) and \(-16 + 4\) in the denominator to get \(-12\). Now, the expression turns into \(\frac{-6}{-12}\).
03

Simplify the fraction

Finally, simplify the fraction \(\frac{-6}{-12}\) to \(0.5\) by dividing the numerator by the denominator. Negative divided by negative gives a positive result.

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Key Concepts

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

Understanding Order of Operations
When solving complex algebraic expressions, it's key to follow the **order of operations**. This ensures every calculation is done consistently and correctly. To remember the proper sequence, use the acronym **PEMDAS**:
  • Parentheses: Solve expressions inside parentheses first.
  • Exponents: Evaluate powers or roots next.
  • Multiplication and Division: Proceed from left to right.
  • Addition and Subtraction: Complete these in the end, also from left to right.
In the given exercise, the expression inside the parentheses **(7-9)** and **(-9-7)** were solved first. This is step #1 of the order. By breaking the problem into these smaller tasks, calculations become much simpler. Always remember to execute operations in brackets first to avoid confusion and mistakes, resulting in an accurate final result.
Mastering Fractions Simplification
Simplifying fractions is about making them as simple as possible, which often involves reducing both the numerator and the denominator by their greatest common divisor. We aim to find more manageable numbers that communicate the same value.
In this exercise, we ended with the fraction \(-\frac{6}{-12}\), which can be simplified.
  • Find the greatest common divisor of both numbers, which in this case, is 6.
  • Divide the numerator and denominator by 6, simplifying it to \(\frac{1}{2}\).
  • Since both numerator and denominator were negative, dividing them results in a positive fraction.
Simplifying fractions not only makes calculations easier but also provides a more elegant expression of the solution.
Dealing with Negative Numbers
Negative numbers can be intimidating, but knowing how they interact during arithmetic operations helps alleviate the fear. Here are some quick tips:
  • When you add a negative number, think of it as subtracting an equivalent positive number.
  • Subtracting a negative number results in an addition operation.
  • Multiplying or dividing two negative numbers results in a positive number.
  • If a positive number is multiplied or divided by a negative, the result is negative.
In the expression, operations with negative numbers occurred in both the numerator and denominator. Understanding that these operations reflect specific rules simplifies calculations and prevents errors. Calculations such as \(-4 + (-2)\) and \(-16 + 4\) showed how knowing the rules helps in solving the task correctly.

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