Problem 79 - Algebra for College Students

Question

Simplify each numerical expression. $$\frac{-6+24}{-3}+\frac{-7}{-6-1}$$

Step-by-Step Solution

Answer

-5

Step 1: Simplify the First Fraction

Start by simplifying the numerator of the first fraction: \(-6 + 24 = 18\). Then divide by the denominator: \(\frac{18}{-3} = -6\).

Step 2: Simplify the Second Fraction

For the second fraction, simplify the denominator first:\(-6 - 1 = -7\). Now divide the numerator by the simplified denominator:\(\frac{-7}{-7} = 1\).

Step 3: Combine the Simplified Values

Add the results from Steps 1 and 2 together:\(-6 + 1 = -5\).

Key Concepts

Simplifying Fractions

Simplifying fractions means breaking down a fraction to its simplest form. Let's think of a fraction as a way to show how many parts of a whole we have. It's made up of two numbers, the numerator and the denominator.

• The numerator is the number on top. It tells us how many parts we have.
• The denominator is the number on the bottom. It shows how many equal parts the whole is divided into.

To simplify a fraction, we need to find the greatest common factor (GCF) of both the numerator and the denominator and then divide them by this number. The goal is to make the fraction as simple as possible.
In the exercise, the first step was to simplify \( \frac{-6+24}{-3} \). Simplifying the expression in the numerator gives 18. We then divide 18 by -3 to make it simpler, resulting in -6. This process makes the fraction easier to work with.

Numerator and Denominator

Every fraction has two main parts: the numerator and the denominator. Understanding these parts is crucial when working with fractions.

• The numerator is the top number of the fraction, and it indicates how many parts of the whole are being considered.
• The denominator, being the bottom number, tells you into how many equal parts the whole is divided.

For example, in the fraction \( \frac{18}{-3} \), 18 is the numerator, which means we have 18 parts. The -3 is the denominator, dividing that quantity into negative portions. That's why when simplifying, it becomes important to manage both numbers correctly. Often, if you're simplifying or performing operations, having a good grasp of these components is needed to adjust the fractions properly.

Addition and Subtraction of Integers

When adding or subtracting integers, it helps to think of them as amounts having direction. Positive numbers are like moving forward, and negative numbers are like moving backward.

• When you add a positive and a negative number, you are essentially finding their difference.
• When both numbers are negative or both are positive, you simply add their absolute values and keep the common sign.

In the exercise, after simplifying both fractions, we performed integer addition: \( -6 + 1 \). Here, -6 is a negative integer, and 1 is a positive integer, leading to a subtraction process that results in -5. By understanding that \( -6 + 1 = -5 \), we see how integer operations help in simplifying numerical expressions.