Problem 11 - Precalculus

Question

Evaluate each expression. $$||-8|+|-9 |$$

Step-by-Step Solution

Answer

The value of the expression is 17.

Step 1: Evaluate the Inner Absolute Values

The expression involves absolute values, so we first evaluate each inner absolute value separately:1. Calculate \(|-8|\): Since the absolute value of a number is its distance from zero, \(|-8| = 8\).2. Calculate \(|-9|\): Similarly, \(|-9| = 9\).

Step 2: Add the Evaluated Absolute Values

Now that we have evaluated the inner absolute values, add them together:\[(|{-8}| + |{-9}|) = 8 + 9 = 17\]

Step 3: Evaluate the Outer Absolute Value

Finally, we need to evaluate the absolute value of the sum from Step 2:Since 17 is already positive, \(|17| = 17\).

Key Concepts

Evaluating Expressions

When you hear about evaluating expressions, it simply means substituting any operations or mathematical processes within the expression to get a final value. For example, when you encounter an expression like \(||-8|+|-9|\), your goal is to perform all necessary calculations to find out the value that this expression represents.

In our exercise, we begin by evaluating what's inside those absolute value symbols because they determine our next steps. It's sort of like peeling an onion - we tackle it layer by layer until we reach the core: the final numeric result.

Inner Absolute Values

The concept of absolute value is central to many expressions. Absolute value, represented by the vertical bars \(|x|\), asks us to consider only the distance of a number from zero, ignoring its sign.

• An inner absolute value such as \(|-8|\) prompts us to think: How far is \(-8\) from zero? Answer: 8 places. So, \(|-8| = 8\).
• Similarly, \(|-9|\) translates to 9, as \(-9\)'s distance from zero is 9.

If you ever see an expression nested in layers of absolute values, you tackle them inward-out like an onion. Once we calculate each inner absolute value, we carry out any other remaining operations. In this case, it's the addition: so you get \(8 + 9 = 17\).

Distance from Zero

Understanding absolute values as distances from zero is a fundamental math concept. It helps simplify how we deal with negative and positive numbers.

When a number is negative, like \-8\ or \-9\, its absolute value converts it to a positive distance. So when you visualize \-8\ on a number line as being 8 units from zero, you come to appreciate how absolute value works as a distance measure.

• Negative numbers transform into their positive counterparts by this distance rule.
• Positive numbers remain as they are when calculating distance from zero, as their signs already denote their distance directly.
• This concept allows us to stick with positive values, which simplifies complex calculations.

Therefore, our result of \(17\) remains unchanged during the outer absolute evaluation since it's a positive value, hence the distance to zero is itself: \(17\). Understanding these basics can empower you to handle even trickier mathematical expressions in the future.