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Chapter 9: Problem 75

Evaluate the following. See Sections 1.5 and 1.6 $$ -(-6)-|-10| $$

Short Answer

Expert verified
The result is -4.

Step by step solution

01

Evaluate the Innermost Negative

The expression contains a double negative: \[-(-6)\]. A double negative turns into a positive, so \[-(-6) = 6\].
02

Evaluate the Absolute Value

Find the absolute value, which is denoted by the vertical bars and represents the distance from zero without considering direction: \[|-10|\]. The absolute value of \(-10\) is \(10\) because it is 10 units away from zero.
03

Subtract the Absolute Value from the Result

Combine the results from the previous steps into the full expression: \[6 - 10\]. Subtract 10 from 6 to evaluate: \[6 - 10 = -4\].

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Key Concepts

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

Understanding Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that both positive and negative numbers share the same absolute value, as distance cannot be negative. To find the absolute value of \(-10\), you simply ignore the negative sign. Thus, \(|-10| = 10\). Absolute value helps in expressions where we only care about the size of the number, not whether it's negative or positive.

Key points about absolute value:
  • Always results in a non-negative number.
  • Indicates magnitude or "size".
Handling Double Negatives
In mathematics, double negatives can occur when two negative signs are placed consecutively. This changes the direction twice, canceling out the negativity, thus resulting in a positive number. Consider the expression \(-(-6)\). The first negative is switched by the presence of the second, turning it into \(6\). Double negatives often arise in subtraction scenarios, and knowing how to simplify them ensures accuracy in calculations.

Important aspects of double negatives:
  • Turns a negative into a positive.
  • Can simplify complex expressions.
Basics of Order of Operations
Order of operations is a set of rules that determines the correct sequence to evaluate a mathematical expression. Known often by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)), these rules ensure consistent results across different calculations.
In the expression \(6 - 10\), we need to adhere to this order. This involves simplifying within parentheses first (if any), then handling exponents, and so forth. For the given exercise, evaluate operations from left to right as there are no parentheses or exponents. This leads directly to subtracting 10 from 6, resulting in \(-4\).

PEMDAS helps by:
  • Providing clear calculation steps.
  • Avoiding confusion in multi-step expressions.

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Most popular questions from this chapter

Solve each compound inequality. Write solutions in interval notation. See Examples I through 8 . $$ -\frac{1}{4}<\frac{6-x}{12}<-\frac{1}{6} $$

Solve each compound inequality for x. See the example below. To solve x-6<3 x<2 x+5, notice that this inequality contains a variable not only in the middle, but also on the left and the right. When this occurs, we solve by rewriting the inequality using the word and. CAN'T COPY THE GRAPH $$ x+3<2 x+1<4 x+6 $$

Graph each inequality. See Examples 2 through 5. $$ x-2 y \geq 3 $$

Solve each compound inequality. Write solutions in interval notation. See Examples 7 and \(8 .\) $$ 3(x-1)<12 \text { or } x+7>10 $$

Evaluate the following. See Sections 1.5 and 1.6 $$ |-4|-(-4)+|-20| $$
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