Problem 81 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(\frac{-6-|-9+5|}{2-(-3)}\)

Step-by-Step Solution

Answer

-2

Step 1: Simplify the Absolute Value

Start by simplifying the absolute value expression inside the numerator: \( |-9 + 5| \). Calculate \( -9 + 5 = -4 \) and then take the absolute value of \( -4 \), which is \( |-4| = 4 \).

Step 2: Substitute Simplified Values

Substitute back the simplified absolute value into the expression: \( \frac{-6 - 4}{2 - (-3)} \).

Step 3: Simplify the Numerator

Simplify the numerator: \( -6 - 4 = -10 \). Now the expression looks like \( \frac{-10}{2 - (-3)} \).

Step 4: Simplify the Denominator

Simplify the denominator: \( 2 - (-3) = 2 + 3 = 5 \). Now the expression becomes \( \frac{-10}{5} \).

Step 5: Divide

Perform the final division: \( \frac{-10}{5} = -2 \).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. It is always non-negative.
For instance, the absolute value of both -4 and 4 is 4, because they are both 4 units away from zero on the number line.
In mathematical notation, we write it as \(|x|\), where x is any real number. For our exercise, we encountered -9 + 5.
Solving it gave us -4. Then, taking the absolute value of -4, we get 4.
Hence, \(|-4| = 4\).
Recap:

• Calculate the expression inside the absolute value
• Take the absolute value of the result

Numerator and Denominator

When dealing with fractions, it's crucial to understand the numerator and denominator:

• The numerator is the top part of the fraction, indicating how many parts we have.


• The denominator is the bottom part, signifying the total number of equal parts the whole is divided into.

In our exercise, we started with the fraction \(\frac{-6-|-9+5|}{2-(-3)}\). After simplifying the absolute value,
it became \(\frac{-6-4}{2-(-3)}\). Here, -6-4 (which simplifies to -10) is the numerator, and 2-(-3) (which simplifies to 5) is the denominator.

Simplifying Expressions

Simplifying expressions involves performing operations to make them easier to work with.
Let's break down our steps:

• Absolute Value: Simplify inside the absolute value first.
• Substitute Values: Replace 4 into the numerator.
• Simplify Numerator: Perform arithmetic operations in the numerator next by calculating -6 - 4 to get -10.
• Simplify Denominator: Then, simplify the denominator; here, changing 2 - (-3) to 2 + 3.

Step-by-step simplification ensures accuracy and makes it easier to reach the final answer.
Always follow the correct order of operations: parentheses, exponents, multiplication and division (from left to right),
and addition and subtraction (from left to right). This is often remembered by PEMDAS (Please Excuse My Dear Aunt Sally).

Division

Division is one of the basic operations in arithmetic. It involves splitting a number into a specified number of equal parts.
In our exercise, after simplifying the numerator and denominator, we obtained the fraction \(\frac{-10}{5}\).
To execute the division, we divide the numerator by the denominator:

• -10 (numerator) divided by 5 (denominator) equals -2.

When dividing, always be careful with signs.
Dividing a negative by a positive results in a negative. Thus, \(\frac{-10}{5} = -2\).
Quick Tips:

• Always simplify the numerator and denominator first, if possible.
• Be mindful of the signs when performing division.