Problem 78 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$|4-\sqrt{2}|-5$$

Step-by-Step Solution

Answer

Question: Simplify the expression \(|4-\sqrt{2}|-5\) and write it without using absolute values. Answer: -1-√2

Step 1: Evaluate the term inside the absolute value

First, we need to determine whether the term \((4-\sqrt{2})\) is positive or negative. Since \(\sqrt{2}\) is approximately 1.41, and 4 > 1.41, we can say \((4-\sqrt{2})\) is positive. So, the absolute value will not change the original term.

Step 2: Remove the absolute value

Now that we know that the term \((4-\sqrt{2})\) is positive, we can remove the absolute value bars and simplify the given expression: $$|(4-\sqrt{2})|-5 = (4-\sqrt{2})-5$$

Step 3: Simplify the expression

Combine the constant terms: $$(4-\sqrt{2})-5 = 4-5-\sqrt{2}$$ $$= -1-\sqrt{2}$$ The simplified expression without using absolute values is: $$-1-\sqrt{2}$$

Key Concepts

Absolute Value Properties

The absolute value of a number reflects its distance from zero on the number line, regardless of direction. For any real number \(x\), the absolute value is denoted as \(|x|\) and is defined by \(|x| = x\) if \(x\) is positive or zero, and \(|x| = -x\) if \(x\) is negative. A key property to remember is that the absolute value of a number will always be non-negative.

When simplifying expressions with absolute values, evaluate the expression within the absolute value bars first. If it’s positive, you can remove the bars without changing the number. If it’s negative, removing the bars means replacing the inner expression with its opposite, or negative. In the exercise, we found that \(4-\sqrt{2}\) is greater than zero, thus the simplification was straightforward by removing the absolute value bars.

Radical Expressions

Radical expressions contain a number under the root symbol, with the square root being the most common. The square root of a number \(x\), denoted by \(\sqrt{x}\), is a value that, when multiplied by itself, equals \(x\). An important property of radicals is that \(\sqrt{x^2} = |x|\), which means the square root of a squared number is the absolute value of the original number, ensuring a non-negative result.

To work with radical expressions effectively, familiarize yourself with the properties of square roots, such as \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) and \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Understanding these properties helps you manipulate and simplify radicals. In the given exercise, the square root of 2 is a non-terminating, non-repeating decimal but is used in simplifying the expression.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This often involves combining like terms, factoring, expanding expressions, and simplifying fractions or radicals. The goal is to make the expression as straightforward as possible while retaining its mathematical integrity.

To simplify algebraic expressions containing absolute values and radicals:

• First, address any operations inside absolute value bars and radicals.
• Remove absolute value bars where appropriate, guided by the sign of the expression within them.
• Combine like terms and simplify any constant and variable expressions.

In the original problem, we combined the steps of evaluating the expression inside the absolute value bars and simplifying terms to reach the final answer \(-1 - \sqrt{2}\). The algebraic simplification process requires attention to detail and a thorough understanding of fundamental algebraic rules to avoid common mistakes.