Problem 26 - Precalculus: Functions and Graphs

Question

Rewrite the expression without using the absolute value symbol, and simplify the result. $$|5-x| \text { if } x>5$$

Step-by-Step Solution

Answer

The simplified expression is \(x - 5\) when \(x > 5\).

Step 1: Understand the Absolute Value Definition

The absolute value of a number, denoted as \(|a|\), is defined as:\[|a| =\begin{cases} a, & \text{if } a \geq 0 \-a, & \text{if } a < 0 \end{cases}\]Given \(|5-x|\), we need to determine whether the expression inside the absolute value is greater than or less than zero when \(x > 5\).

Step 2: Determine the Sign of the Inside Expression

For \(|5-x|\) and \(x > 5\), substitute a value greater than 5, such as 6:\(5 - 6 = -1\).This result is negative, indicating that \(5-x < 0\) when \(x > 5\).

Step 3: Rewrite Without Absolute Value

Since \(5-x < 0\) from Step 2, rewrite the absolute value expression as its negative:\(|5-x| = -(5-x)\)\ when \(x > 5\).

Step 4: Simplify the Expression

Simplify the expression from Step 3:\[-(5-x) = -5 + x = x - 5\]Thus, the expression without the absolute value sign and simplified is \(x-5\) when \(x > 5\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that describe the relationship between two values, showing whether one is greater than, less than, or equal to another. They use symbols such as:

• \(<\) for "less than"
• \(>\) for "greater than"
• \(\leq\) for "less than or equal to"
• \(\geq\) for "greater than or equal to"

Understanding inequalities is crucial when working with absolute values because they help us determine the range of values for which an expression might change its sign.
In our exercise, the inequality \(x > 5\) informs us that, inside the absolute value expression \(|5-x|\), the term \(5-x\) becomes negative. Hence, knowing inequalities allows us to determine the appropriate re-writing and simplification of expressions involving absolute values.

Exploring Piecewise Functions

Piecewise functions are functions composed of multiple sub-functions, each defined over a distinct part of the domain. These functions are crucial for understanding absolute value expressions, as absolute values can often be interpreted as piecewise functions.
The definition of an absolute value is inherently piecewise since it differentiates the behavior of the expression based on whether the input is non-negative or negative.
For example:

• If the input is non-negative, the absolute value function outputs the original number.
• If the input is negative, it outputs the negation of that number.

For the expression \(|5-x|\), when \(x > 5\), the input \(5-x\) is negative, so the absolute value function effectively behaves as the piece \(-(5-x)\), which simplifies to \(x - 5\).
Understanding this piecewise mentality helps you unpack and work through complex functions that appear in different mathematical contexts.

Expression Simplification Techniques

Expression simplification is the process of rewriting complex expressions in a simpler or more understandable form. It involves combining like terms, reducing constants, or applying algebraic properties.
Simplification can make solving equations or comparing expressions much easier and is an important skill in algebra and calculus.
In this exercise, after rewriting the absolute value expression \(|5-x|\) as \(-(5-x)\) when \(x > 5\), we apply simplification:

• Distribute the negative sign: \(-(5-x) = -5 + x\)
• Rearrange and combine terms, yielding the final simplified form: \(x - 5\)

These steps illustrate how expression simplification involves basic operations and a clear understanding of algebraic rules to achieve a neat and concise result.