Problem 62 - Algebra and Trigonometry

Question

\(61-66=\) Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }|\sqrt{5}-5|} & {\text { (b) }|10-\pi|}\end{array} $$

Step-by-Step Solution

Answer

(a) 2.764, (b) 6.85841

Step 1: Understand Absolute Value Function

The absolute value function, denoted by the symbol \(|x|\), gives the distance of a number from 0 on the number line, always yielding a non-negative result. Thus, \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\).

Step 2: Evaluate Expression (a)

Expression (a) is \(|\sqrt{5} - 5|\). First, calculate \(\sqrt{5} \approx 2.236\). Then, compute \(\sqrt{5} - 5 = 2.236 - 5 \approx -2.764\). Because the result is negative, we take its absolute value: \(|\sqrt{5} - 5| = |-2.764| = 2.764\).

Step 3: Evaluate Expression (b)

Expression (b) is \(|10 - \pi|\). The value of \(\pi \approx 3.14159\). Calculate \(10 - \pi = 10 - 3.14159 \approx 6.85841\). Since this result is positive, the absolute value doesn't change it: \(|10 - \pi| = 6.85841\).

Step 4: Present Results for Each Expression

Summarize the calculations: (a) \(|\sqrt{5} - 5| = 2.764\)(b) \(|10 - \pi| = 6.85841\)

Key Concepts

Distance on Number Line

The number line is a visual tool that helps us understand the concept of distance between numbers. In essence, the distance between any two numbers on a number line is the absolute difference between them. When we talk about absolute value in math, we are referring to how far a number is from zero, regardless of its direction. For example,

• The absolute value of -4, denoted as \(|-4|\), is 4.
• This can be understood as a distance because \(-4\) is 4 units away from zero on the number line.

Applying this principle, the distance of any negative number to zero is turned into a positive number because distance can never be negative. Thus, when evaluating expressions using absolute values, you're essentially finding how much ground you cover on the number line between two points. This is crucial for understanding evaluations like \(|\sqrt{5} - 5|\), where we are measuring how far \sqrt{5}\ is from 5 on a number line.

Evaluation of Expressions

Evaluating expressions involves simplifying them to find their value, which follows rules of arithmetic. When given an expression like \(|\sqrt{5} - 5|\), the first step is to simplify the operation within the absolute value. Here's how it breaks down:

• Compute \sqrt{5}\ which approximately equals 2.236.
• Subtract this from 5 to get \(-2.764\).

Since this result is negative, we employ the absolute value function, flipping the sign. Therefore, \(|\sqrt{5} - 5| = 2.764\), turning the negative outcome into a positive distance or evaluation result. By following these similar steps, you can always effectively evaluate expressions that include whole numbers, fractions, or constants like \pi\.

Square Root

The square root process is about finding a number which, when multiplied by itself, results in the given number. The symbol for square root is \(\sqrt{}\)\. Understanding this helps in evaluating expressions like \(|\sqrt{5} - 5|\). When we look for \(\sqrt{5}\)\, we seek a number \(\approx 2.236\)\ because \(2.236 \times 2.236 \approx 5\).

• Square roots guide us to see the sense of approximation, as not every square root results in a whole number.
• Thus, it is common to approximate square roots to make further calculations possible.

This concept is key when square roots appear in other expressions or equations, as square roots help us resolve parts of calculations that are more complex than simple arithmetic.

Value of Pi

The value of \(\pi\), denoted as approximately 3.14159, is a constant that frequently appears in mathematics, particularly in calculations relating to circles. \pi\ represents the ratio of the circumference of a circle to its diameter, which is why it's an integral part of many geometry-related expressions and problems.

• In terms of its application in evaluating expressions, \pi\ is often used as-is for calculations, like in \(|10 - \pi|\). \
• This particular usage is often seen in mathematical problems where the measurement of angles or circles is needed.

With an understanding of \pi\, evaluating expressions where \pi\ appears involves common operations where you substitute the approximate value of \pi\, leading to the evaluation of expressions like \(|10 - \pi| = 6.85841\). Understanding such constants ensures accurate expression evaluations.