Problem 109 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 65 \div(-5)+(-28) \div(-7) $$

Step-by-Step Solution

Answer

-9

Step 1: Divide 65 by -5

First, calculate the division of 65 by -5. \[ 65 \div (-5) = -13 \]

Step 2: Divide -28 by -7

Next, calculate the division of -28 by -7. \[ (-28) \div (-7) = 4 \]

Step 3: Add the Results

Finally, add the results from the previous steps together. \[ -13 + 4 = -9 \]

Key Concepts

Division of Integers

In mathematics, division of integers is a fundamental operation that involves dividing one integer by another. When dividing integers, it's important to understand the rules that govern the signs of the numbers involved.

• When both the dividend and divisor have the same signs (both positive or both negative), the result is positive.
• When the signs of the dividend and the divisor are different, the result is negative.

Let's look at an example:
If we have 65 divided by -5, the divisor is negative while the dividend is positive. According to the rule, the result will be negative: \( 65 \div (-5) = -13 \)
Similarly, for -28 divided by -7, both integers are negative, so the result will be positive: \( (-28) \div (-7) = 4 \)
Understanding these principles is key to mastering integer division and helps in solving more complex problems.

Negative Numbers

Negative numbers are numbers less than zero, and they are usually denoted with a minus sign (-). They play a crucial role in mathematics and are often encountered in various arithmetic operations such as addition, subtraction, multiplication, and division.

• Negative of a negative number results in a positive number. For example, \( -(-7) = 7 \).
• Adding a negative number is the same as subtracting its positive counterpart. For example, \( 5 + (-3) = 5 - 3 = 2 \).

Negative numbers are essential in real-life contexts such as representing debts, temperatures below freezing, and altitudes below sea level. In the given exercise, we encountered negative numbers in division operations, and understanding how to handle them correctly is essential in simplifying expressions involving negative values.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent and accurate results. The common acronym to remember the order is PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• Parentheses: Solve expressions inside parentheses first.
• Exponents: Calculate powers or square roots next.
• Multiplication and Division: Carry out these operations next, from left to right.
• Addition and Subtraction: Lastly, carry out these operations, again from left to right.

In our initial exercise, operations were carried out in the correct order to simplify the expression:
\( 65 \div (-5) + (-28) \div (-7) \)
First, we performed the divisions: \( 65 \div (-5) = -13 \) and \( (-28) \div (-7) = 4 \).
Then, we added the results: \( -13 + 4 = -9 \).
By following the order of operations, we arrived at the correct simplified expression.