Problem 83 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The numerical expression simplifies to -5.

Step 1: Simplify the numerator of the first fraction

Look at the fraction \( \frac{-6+24}{-3} \). Simplify the numerator by adding \( -6 \) and \( 24 \), which gives \( 18 \). Therefore, the expression simplifies to \( \frac{18}{-3} \).

Step 2: Calculate the first fraction

Divide the simplified numerator by the denominator: \( \frac{18}{-3} = -6 \). So, the first part of the expression is \( -6 \).

Step 3: Simplify the denominator of the second fraction

Look at the fraction \( \frac{-7}{-6-1} \). Simplify the denominator by adding \( -6 \) and \( -1 \), which gives \( -7 \). Therefore, the expression becomes \( \frac{-7}{-7} \).

Step 4: Calculate the second fraction

Divide \( -7 \) by \( -7 \), which is \( 1 \). So, the second part of the expression is \( 1 \).

Step 5: Combine the results of the expressions

Take the results from the first and second parts: \( -6 \) and \( 1 \). Add them together: \( -6 + 1 = -5 \).

Step 6: Final Step: Write the simplified expression

The entire expression \( \frac{-6+24}{-3} + \frac{-7}{-6-1} \) simplifies to \( -5 \).

Key Concepts

Fractions

Fractions are a way to represent a part of a whole or a division between two quantities. They consist of two main parts: the numerator, which sits on top, and the denominator, which is below the line. For instance, in the fraction \( \frac{3}{4} \), \( 3 \) is the numerator and \( 4 \) is the denominator.
Understanding fractions is crucial when simplifying numerical expressions, as they are often involved in breaking down more complex calculations. In our original exercise, the fraction plays a key role in organizing operations involving integers.

Numerator and Denominator

The numerator and denominator are essential components of a fraction. They help determine the portion of a whole being represented or divided.

• The numerator: It represents how many parts of a whole are taken. In the fraction \( \frac{-6+24}{-3} \), after simplifying \(-6 + 24\), our numerator is \(18\).
• The denominator: It indicates how many equal parts the whole is divided into. In the same fraction \( \frac{18}{-3} \), \(-3\) is the denominator.

By fully understanding the roles of these two components, you can simplify and calculate fractions effectively. When a numerator equals the denominator, like in \( \frac{-7}{-7} \), the fraction reduces to \(1\). This understanding simplifies complex expressions significantly.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are fundamental to manipulating fractions and simplifying expressions. When dealing with fractions, these operations often require keeping an eye on both numerators and denominators.

• Addition/Subtraction: Simplify each part of the expression separately, as seen when \(-6\) and \(24\) are combined to simply the numerator.
• Multiplication/Division: In the original problem, division occurs between the numerator and the denominator, such as \( \frac{18}{-3} = -6 \), an example of simple division.

Understanding these operations allows you to manipulate fractions more effectively, paving the way to a simplified result such as the final value of \(-5\).

Step-by-Step Solutions

Breaking down a seemingly complex problem into manageable parts allows us to find solutions more easily and accurately. When simplifying numerical expressions, a step-by-step approach ensures that each component is tackled methodically.
Here’s a quick reminder of the process in our example:

• Simplify the Numerator: Start by resolving operations in the numerator, like \(-6 + 24\) resulting in \(18\).
• Calculate the first fraction: Divide the simplified numerator by the denominator to get \(-6\).
• Simplify the Denominator: Address the denominator operations such as \(-6 - 1\) to get \(-7\).
• Calculate the second fraction: \( \frac{-7}{-7} \) simplifies to \(1\).
• Combine the results: Adding the results, \(-6 + 1 = -5\), gives the overall simplified expression.

Using step-by-step solutions can help clarify each process and provides a clear path to the final answer, ensuring nothing is overlooked.