Problem 84 - Intermediate Algebra

Question

Simplify each numerical expression. $$ \frac{-12+20}{-4}+\frac{-7-11}{-9} $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Simplify the Numerators

Start by simplifying the numerators in both fractions separately. For the first fraction, calculate \(-12+20\): \[ -12 + 20 = 8 \]. For the second fraction, calculate \(-7-11\): \[ -7 - 11 = -18 \].

Step 2: Calculate Each Fraction

Now, compute the value of each simplified fraction:1. \(\frac{8}{-4}\): \[ \frac{8}{-4} = -2 \].2. \(\frac{-18}{-9}\): \[ \frac{-18}{-9} = 2 \].

Step 3: Add The Results

Add the results of the two fractions from Step 2:\(-2 + 2 = 0\).

Key Concepts

Simplification

Simplification is a critical mathematical process that helps make complex expressions easier to understand and solve. In algebraic fractions, simplification often involves reducing fractions to their simplest form or handling the arithmetic operations like addition or subtraction within the fraction itself. In the given exercise, each fraction was simplified by performing arithmetic operations on the numerators. This process involves several clear steps:

• Identify which mathematical operations are present in the fraction's numerator.
• Perform these operations, ensuring you correctly apply the rules of arithmetic, such as dealing with negative numbers.
• Simplify the fraction by dividing the numerator by the denominator if possible.

By simplifying fractions, we can easily see relationships and outcomes, making further calculations much more straightforward.

Numerical Expression

A numerical expression in mathematics is a combination of numbers and operations that represent a value. It does not have a variable component, which distinguishes it from algebraic expressions. In terms of fractions, numerical expressions use arithmetic operations like addition, subtraction, multiplication, or division to derive values.
In our exercise, the numerical expression given is a fraction with operations in the numerator that needed to be simplified:

• The expression inside the first fraction was \(-12 + 20\), which required addition of a negative number and a positive number.
• The expression inside the second fraction was \(-7 - 11\), involving subtraction of negative numbers.

Once these expressions were resolved, the numerators \(8\) and \(-18\) respectively, further simplified the fractions. Numerical expressions form the backbone of simplifying algebraic fractions, by allowing us to evaluate and simplify step by step.

Fractions

Fractions are one of the fundamental concepts in mathematics, representing parts of a whole. They consist of a numerator and a denominator, where the numerator is the part and the denominator is the whole.
For our given problem, understanding fractions involved several steps to simplify the numerical expressions and solve the problem:

• Firstly, simplify the expression in the numerator before attempting any division.
• With the simplified numerators, calculate each resulting fraction, \(\frac{8}{-4}\) and \(\frac{-18}{-9}\).
• Understanding the meaning of a negative sign with a fraction is crucial. The negative symbol indicates the direction, or more simply, which part is considered negative.

Fractions offer a way to operate on non-integer values or among values that represent proportions, requiring clear comprehension to simplify expressions accurately and achieve correct results.