Problem 17 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ |5-8 \cdot 7| $$

Step-by-Step Solution

Answer

The value is 51.

Step 1: Evaluate the Expression Inside the Absolute Value

The expression is \(5 - 8 \cdot 7\). Follow order of operations, or PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).First, handle the multiplication: \(8 \cdot 7 = 56\).Now, substitute back into the expression: \(5 - 56\).

Step 2: Calculate the Subtraction

Compute the subtraction in the expression: \(5 - 56 = -51\).

Step 3: Apply Absolute Value

The absolute value function transforms negative numbers into their positive counterparts. Thus, the absolute value of \(-51\) is \(|-51| = 51\).

Key Concepts

Absolute Value

Absolute value is a way to express a number's distance from zero on the number line, without considering direction. It is always non-negative. Absolute values are denoted using vertical bars; for instance, \(|x|\) represents the absolute value of \(x\).

Let's say we have a number, whether positive or negative, the absolute value essentially translates it to its positive counterpart.

• If \(x > 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\). This means we are effectively "removing" the negative sign.
• If \(x = 0\), then \(|x| = 0\).

For example, if you have the number -3, its absolute value is 3 because it is 3 units away from zero. In the exercise provided, applying it yields \(|-51| = 51\). It makes no difference whether the initial number was negative; absolute value ensures a positive result.

Order of Operations

Understanding the order of operations is crucial when evaluating arithmetic expressions. In mathematics, operations must be performed in a specific sequence to achieve the correct result. This sequence is often remembered by the acronym PEMDAS:

• Parentheses: Solve anything inside parentheses first.
• Exponents: Resolve powers or roots second.
• Multiplication and Division: These operations are solved third, moving from left to right.
• Addition and Subtraction: Finally, these are also performed from left to right.

To illustrate, let's take the expression \(5 - 8 \cdot 7\). We need to tackle multiplication before subtraction according to PEMDAS. That's why \(8 \cdot 7\) is computed first to get 56, and then it replaces the expression to get \(5 - 56\).

By strictly following this order, we avoid errors in calculations that could result from performing operations out of sequence. This systematic approach ensures consistent and accurate evaluation of expressions.

Arithmetic Expressions

An arithmetic expression is a mathematical phrase involving numbers and operation symbols. These operation symbols can include addition \(+\), subtraction \(-\), multiplication \(\cdot\), and division \(\div\). Each expression has at least one operation involved.

When evaluating arithmetic expressions, it's important to determine which operations to perform first. In the expression \(5 - 8 \cdot 7\), there are both subtraction and multiplication involved. Denoted operations can be sorted by rules like the order of operations discussed earlier.

• First, the multiplication is addressed: \(8 \cdot 7 = 56\).
• Then, the result is used in the subtraction: \(5 - 56 = -51\).

This careful and organized manner of breaking down and evaluating an expression is key to successfully solving problems. Always remember to apply the necessary operations systematically, especially if expressions become more complex.