Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ 4+|-4| $$

Step-by-Step Solution

Answer

The given expression is \( 4 + |-4| \). Evaluating the absolute value function, we get \( |-4| = 4 \). Thus, the expression becomes \( 4 + 4 \), which equals 8.

Step 1: Evaluate the absolute value function

The expression contains an absolute value function: \( |-4| \). The absolute value of a number is its distance from zero on the number scale. It is always a non-negative number because distance is always non-negative. The absolute value of -4 is 4 because it is 4 units away from zero on the number scale. So we can rewrite the expression as: \( 4 + 4 \).

Step 2: Perform the addition

Now we can just add the numbers in the expression: \( 4 + 4 = 8 \). The final result of the expression is 8.

Key Concepts

Expression Evaluation

Evaluating an expression involves identifying various parts of the expression and simplifying them. In mathematical expressions, we often encounter various operators and numbers, such as addition, subtraction, multiplication, and division. However, some expressions also include functions like the absolute value.

Expression evaluation is the process of simplifying these operations to obtain a single number as the result. Start by addressing any special operations, such as absolutes, to convert the expression step by step into a solvable formula.

• Identify functions or operations like absolute values within the expression.
• Simplify each component of the expression, step by step.
• Perform calculations to achieve a single numerical result.

Breaking down an expression into smaller parts makes it easier to manage and solve.

Addition

Addition is one of the most fundamental mathematical operations. It involves combining numbers to find their total or sum. In expressions, once all components are simplified, you often need to perform addition to reach a final result.

In the expression evaluated here, after simplifying the absolute value, what remains is a simple arithmetic addition. Here’s a reminder of key aspects of addition:

• Adding two positive numbers results in a larger positive number.
• The order in which numbers are added does not affect the sum (commutative property).
• Addition can be visualized as moving right on a number line.

Addition simplifies to a straightforward calculation once any complex operations in an expression are resolved.

Distance on Number Scale

Understanding the concept of distance on a number scale is crucial for comprehending absolute values. Absolute values are related to the distance concept because they measure how far a number is from zero, without considering direction.

Here’s how distance works on a number scale:

• Any positive number has an absolute value equal to itself.
• For negative numbers, the absolute value is the positive counterpart of the number.
• The absolute distance is the number of units from zero, which is always a non-negative number.

Considering absolute value as distance helps simplify expressions by converting negative values to positive ones, aiding in further arithmetic operations.