Problem 77 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$|3-\pi|+3$$

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is \(|3-\pi|+3=3.1416\).

Step 1: Simplify the expression inside the absolute value

We need to simplify \(3-\pi\). Since \(\pi \approx 3.1416\), we have: $$3-\pi=-0.1416$$

Step 2: Evaluate the absolute value

Now we need to find the absolute value of \(-0.1416\). As the absolute value of a number is the non-negative value of the number, we get: $$|-0.1416|=0.1416$$

Step 3: Add the result to 3

Finally, we need to add the result from the previous step to 3: $$0.1416+3=3.1416$$ Therefore, the simplification of the given expression is: $$|3-\pi|+3=3.1416$$

Key Concepts

Absolute Value

The absolute value of a number is like taking away any negative sign it might have. Imagine it as the distance of a number from zero on a number line. Whether the number is positive or negative, its absolute value is always positive.
- For example, the absolute value of \(-5\) is \(|-5| = 5\). - The absolute value of \(7\) is \(|7| = 7\). So, when we deal with an expression inside an absolute value like \(3 - \pi\), we first compute the expression. Since \(\pi \) (approximately 3.1416) is slightly bigger than 3, \(3 - \pi\) becomes negative.
But with absolute value, negatives disappear, turning \(-0.1416\) into \(0.1416\). This positive result is what we use moving forward.

Simplifying Expressions

Simplifying expressions can be likened to tidying up a room. We combine like terms and perform arithmetic to make the expressions as straightforward as possible.
In the exercise, we started with \(|3 - \pi| + 3\). Here’s how we progress:

• Simplify inside the absolute value first, finding \(3 - \pi = -0.1416\).
• Apply the absolute value: \(|-0.1416| = 0.1416\).
• Finally, add this result to 3: \(0.1416 + 3 = 3.1416\).

This step-by-step helps ensure every bit of the expression is simplified correctly and clearly without extra clutter.

Real Numbers

Real numbers include all the numbers you can think of on the number line, whether they are whole like 2, fractions like \(\frac{1}{2}\), or irrational like \(\pi\). When working with real numbers:

• They can be positive, negative, or zero.
• They encompass both rational numbers (e.g., 3, 0.75) and irrational numbers (e.g., \(\pi\), \(\sqrt{2}\)).

In our exercise, \(\pi\), which is around 3.1416, is an irrational number.
When subtracting \(\pi\) from 3, we stay within the family of real numbers, producing the decimal \(-0.1416\). Using real numbers allows us to perform addition, subtraction, and other standard operations while remaining consistent no matter the kind of number.