Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-32: 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 expression \(|5-x|\) simplifies to \(x-5\) for \(x > 5\).

Step 1: Understanding Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. Therefore, \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\).

Step 2: Evaluating Expression for Given Condition

Given the expression \(|5-x|\) and the condition \(x > 5\), we need to determine whether \((5-x)\) is positive or negative. Since \(x > 5\), \(5-x < 0\). Therefore, the expression inside the absolute value is negative.

Step 3: Rewrite Absolute Value Expression

Since \(5-x\) is negative for \(x > 5\), the expression \(|5-x|\) simplifies to \(-(5-x)\), which simplifies further to \(x-5\).

Step 4: Simplification

After removing the absolute value symbols and rewriting, the expression becomes \(x-5\). This is already in its simplest form.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions help us represent situations in a concise form. For example, the expression \(5-x\) involves a variable "\(x\)" and a constant "5". Variables like "\(x\)" can take on different values, which makes algebraic expressions extremely useful in solving real-world problems. By using algebraic expressions, we can model complex relationships and make calculations that would otherwise be cumbersome.

Inequalities

Inequalities express the relationship between expressions that are not equal. They use symbols such as \(>\), \(<\), \(\geq\), and \(\leq\) to show this relationship. In the given exercise, the inequality \(x > 5\) indicates that \(x\) is greater than 5. This kind of condition is crucial when determining how to handle absolute values. Since \(x\) is greater than 5, the difference \(5-x\) turns out to be less than zero, which helps us decide how to simplify the absolute value expression. Understanding how to manipulate and interpret inequalities is vital when solving algebraic problems that involve absolute values.

Simplification

Simplification is the process of reducing an expression to its simplest form. In this specific problem, we start by analyzing the expression \(|5-x|\). Knowing that \(5-x\) is negative when \(x > 5\), we rewrite the absolute value \(|5-x|\) as \(-(5-x)\). Simplifying further involves removing the parentheses and changing signs, leading to \(x-5\). Simplification helps when solving equations and inequalities, as it makes calculations easier and results more understandable. It provides a clear and concise way to express answers.

Number Line Distance

The concept of number line distance is at the core of understanding absolute values. The absolute value represents the distance a number is from zero on the number line, regardless of direction. Unlike regular numbers, absolute values are always non-negative. In this exercise, the expression \(|5-x|\) measures the distance of the value \(5-x\) from zero. Since distance cannot be negative, if \(5-x\) is negative due to \(x > 5\), the absolute value reverses the sign, ensuring the expression is positive. Grasping the idea of number line distance helps in visualizing how absolute value functions and assists in correctly rewriting and simplifying expressions based on given conditions.