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Chapter 3: Problem 387

In the following exercises, simplify. $$ -(-55) \text { and }-|-55| $$

Short Answer

Expert verified
55 and -55

Step by step solution

01

Simplify the first expression \(-(-55)\)

The first part of the exercise is to simplify \(-(-55)\). When dealing with a double negative, the two negatives cancel each other out. Therefore, we are left with: \(-(-55) = 55\)
02

Simplify the second expression \(-| -55 |\)

The second part of the exercise is to simplify \(-|-55|\). The absolute value of a number is always positive, so \(|-55| = 55\). Then we need to take the negative sign of the absolute value result: \(-| -55 | = -55\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double Negatives
Double negatives in mathematics work similarly to how they do in language. When you see two negative signs one after the other, like in \(-(-55)\), they cancel each other out. Think of it as flipping twice. The first flip turns \(-55\) into positive \(55\), and the second flip doesn't change it since it's already positive. So \(-(-55)\) simplifies straight to \(+55\). This concept is especially important when solving equations as it helps to simplify the expressions quickly.
Absolute Value
The absolute value of a number, represented by vertical bars like this \(|-55|\), is the distance of that number from zero on a number line, without considering direction. It turns all numbers into non-negative values. For instance, the absolute value of \(-55\) is \(+55\) because the distance between -55 and 0, regardless of direction, is 55 units. In our exercise, \(-| -55 |\) becomes \(-55\) after taking the absolute value.
Simplifying Expressions
Simplifying expressions is about making them as straightforward as possible by combining like terms and applying mathematical rules. Here's a quick guide:
  • Combine like terms: This means adding or subtracting terms that have the same variables raised to the same power.
  • Use rules of arithmetic: Apply operations such as addition, subtraction, multiplication, and division in the correct sequence.
  • Reduce fractions to their simplest form: If the numerator and the denominator have a common factor, divide both by this factor.
  • Handle exponents: Apply the rules for multiplying and dividing powers, and for handling negative or zero exponents.
  • Remove parentheses: Expand expressions by distributing multiplication over addition/subtraction.
  • Pay attention to signs: Remember to follow the rules for adding, subtracting, multiplying, and dividing positive and negative numbers.
Using these steps systematically makes seemingly complex problems much simpler. In our exercise, applying the rules of double negatives and absolute values sequentially helped us to simplify the expressions efficiently.

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