Problem 76 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 45-5|1-8| $$

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Evaluate Inside the Absolute Value

The expression inside the absolute value is \(1 - 8\). Since simple arithmetic is involved, subtract 8 from 1 to get -7. Therefore, \(|1 - 8| = |-7|\).

Step 2: Find the Absolute Value

The absolute value of a number is the non-negative value. Thus, \(|-7| = 7\).

Step 3: Subtract from 45

Now, replace \(|1-8|\) in the original expression with 7. The expression becomes \(45 - 5 imes 7\). According to order of operations, multiply first: \(5 imes 7 = 35\).

Step 4: Complete the Subtraction

Subtract the product from 45: \(45 - 35 = 10\).

Key Concepts

Order of Operations

When evaluating mathematical expressions, following the correct order of operations is crucial to arriving at the right answer. This set of rules dictates the sequence in which operations should be performed:

• First, handle any operations inside parentheses or brackets.
• Next, evaluate expressions with exponents or roots.
• Then, perform any multiplication and division, working from left to right.
• Finally, execute any addition and subtraction, also from left to right.

In our example, after evaluating the absolute value, we proceed to multiplication before subtraction. This is in line with the PEMDAS/BODMAS rule (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Beware that ignoring this sequence might lead to incorrect results. Thus, always remain diligent in following the order to ensure accuracy in your math problems.

Subtracting Integers

Subtracting integers involves finding the difference between numbers and can sometimes be tricky due to the involvement of positive and negative values. A helpful way to think about subtraction is as the addition of a negative. For instance:

• When subtracting a positive number, it's like taking away something.
• When subtracting a negative number, it is equivalent to adding its positive counterpart.

In the exercise, we start with the integer subtraction within the absolute value, specifically computing \(1 - 8\). Here, subtracting 8 from 1 is essentially moving 8 units to the left of 1 on a number line, resulting in \(-7\). Understanding these nuances is vital for correctly handling mathematical operations involving integers.

Multiplication

Multiplication is a fundamental arithmetic operation that involves calculating the total of one number taken a certain number of times. It's often viewed as repeated addition. For example, \(5 \times 7\) means the number 5 added together 7 times. In our expression \(5 \times 7 = 35\), we utilize multiplication during the order of operations process. Given its repeated nature, multiplication simplifies and expedites calculations, especially when dealing with larger numbers or multiple factors.
By ensuring precise multiplication before carrying out subtraction in our problem, we achieve the correct total. Proper handling of this operation is vital, as it affects the outcome of the entire mathematical expression.