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Chapter 2: Problem 36

For Exercises \(1-74,\) simplify. $$|21 \div(-3)+12|$$

Short Answer

Expert verified
Answer: 5

Step by step solution

01

Perform Division

First, perform the division operation inside the absolute value: \(21 \div(-3)\) \(21 \div(-3) = -7\) Now the expression becomes: \(|-7 + 12|\)
02

Perform Addition

Next, perform the addition operation inside the absolute value: \(-7 + 12\) \(-7+12 = 5\) Now the expression becomes: \(|5|\)
03

Compute Absolute Value

Finally, we need to compute the absolute value of the result: \(|5|\) As 5 is a positive number, its absolute value is equal to its value: \(|5| = 5\) So, the simplified expression is: $$5$$

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Key Concepts

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

Division in Algebra
Division in algebra is a key mathematical operation. It involves dividing one quantity by another. If we look at the exercise, we divide 21 by -3, expressed as \(21 \div (-3)\). This division leads to \(-7\).
This step is crucial for simplifying algebraic expressions, helping to break down complex problems into simpler parts.
Here are some important points to remember about division in algebra:
  • Sign Rules: When dividing, remember that a positive number divided by a negative number results in a negative number.
  • Inverse Operation: Division is the inverse of multiplication. So, \(a \div b\) is like multiplying \(a\) by the reciprocal of \(b\), which is \(1/b\).
The division operation often sets the stage for the next steps in simplifying the expression.
Absolute Value
The absolute value of a number is its distance from zero on a number line, regardless of direction. In simpler terms, it turns any number into its non-negative version.
Looking at the exercise, after performing division and addition, we have the expression \(|5|\). Since 5 is already positive, its absolute value is 5.
Some key points about absolute value:
  • Non-Negativity: The absolute value of any number is always non-negative. For instance, \(|-3| = 3\) and \(|3| = 3\).
  • Functionality: Think of absolute value as a function that outputs the magnitude of the input without its sign.
Absolute value is often used to simplify expressions, especially those involving negative values.
Order of Operations
The order of operations is a fundamental concept in mathematics that dictates the sequence in which operations should be performed to accurately solve expressions. In our exercise, this principle guides us in simplifying \(|21 \div(-3)+12|\).
Follow these steps to ensure accuracy:
  • Parentheses: Perform operations inside parentheses first. However, our expression uses absolute value bars which function similarly in terms of hierarchy order.
  • Exponents: Calculate any exponents second, though none are present in this case.
  • Multiplication and Division: These are handled next, and from left to right. In the exercise, division was the first step: \(21 \div(-3)\).
  • Addition and Subtraction: Finally, perform addition and subtraction from left to right. Here, \(-7 + 12\) was performed after the division.
Understanding the order of operations ensures expressions are simplified correctly and accurately.

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