Problem 80 - Algebra for College Students

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 Numerator of the First Fraction

Start by simplifying the numerator of the first fraction, which is \(-12 + 20\). Calculate this by adding the two numbers: \(-12 + 20 = 8\).

Step 2: Simplify the First Fraction

Use the simplified numerator from Step 1 to simplify the fraction: \(\frac{8}{-4}\). Divide 8 by -4 to get \(-2\).

Step 3: Simplify the Numerator of the Second Fraction

Now simplify the numerator of the second fraction, which is \(-7 - 11\). Calculate this by adding the two negative numbers: \(-7 - 11 = -18\).

Step 4: Simplify the Second Fraction

Using the simplified numerator from Step 3, simplify the second fraction: \(\frac{-18}{-9}\). Divide -18 by -9 to get \(2\).

Step 5: Add the Simplified Fractions

Add the results from Step 2 and Step 4 together: \(-2 + 2 = 0\). Thus, the original expression simplifies to \(0\).

Key Concepts

Numerical Expressions

Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, or division. They do not contain any variables, which makes them different from algebraic expressions. The goal with a numerical expression is to evaluate or simplify it, which requires understanding and applying arithmetic operations correctly.

For example, in the expression given: \( \frac{-12+20}{-4}+\frac{-7-11}{-9} \), both fractions are themselves numerical expressions that we need to evaluate before summing them up. This involves executing arithmetic operations one step at a time.

Fractions

Fractions are mathematical expressions representing the division of one quantity by another. They consist of two parts: a numerator and a denominator. The numerator is the number above the line, and the denominator is the number below it.

In our exercise, we deal with fractions such as \( \frac{-12+20}{-4} \) and \( \frac{-7-11}{-9} \). Here, we start by simplifying each fraction by performing the arithmetic operations in the numerator first, and then dividing by the denominator.

Understanding fractions as a division operation helps grasp how they are manipulated and simplified.

Simplification

Simplification in mathematics means reducing an expression to its simplest form. This often involves performing operations to break down complex expressions into more manageable parts.

For the first fraction \( \frac{-12+20}{-4} \), simplification means first computing the numerator \(-12 + 20\) to get \(8\). Then, dividing by the denominator \(-4\) results in \(-2\). Similarly, for the second fraction \( \frac{-7-11}{-9} \), simplify the numerator to \(-18\) and divide by \(-9\) to get \(2\).

Simplification makes further calculations easier and helps in understanding the problem's basic structure.

Addition

Addition is one of the fundamental arithmetic operations, representing the combining of quantities. In expressions involving numbers and fractions, after simplification, addition is crucial to derive the entire expression's value.

In this problem, once both fractions are simplified individually to \(-2\) and \(2\), we need to perform addition. So, adding these two numbers gives \(-2 + 2 = 0\).

When working with addition, especially with numbers having different signs, understanding how positive and negative numbers interact is essential.

Division

Division is an arithmetic operation where a number, known as the dividend, is divided by another number, called the divisor. The result is called the quotient. Fractional division is directly represented by fractions, where the numerator is the dividend and the denominator is the divisor.

For instance, \( \frac{8}{-4} \) means that \(8\) is divided by \(-4\), resulting in \(-2\). Similarly, \( \frac{-18}{-9} \) results in \(2\).

Understanding division helps in breaking down complex numerical expressions into simpler components, facilitating easier computation and comprehension.