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Chapter 2: Problem 37

For Exercises \(1-74,\) simplify. $$-|-43+6 \cdot 4|$$

Short Answer

Expert verified
Question: Simplify the expression \(-|-43 + 6 \cdot 4|\). Answer: -19

Step by step solution

01

Simplify the expression inside the absolute value

First we need to simplify the expression inside the absolute value. According to PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, AndAlso Addition and Subtraction from left to right), we need to perform the multiplication operation before addition: $$-|-43+6\cdot 4| = -|-43+24|$$ Now we can add the numbers inside the absolute value: $$-|-43+24| = -|-19|$$
02

Take the absolute value

Now we can compute the absolute value of \(-19\): $$-|-19| = -|19|$$
03

Multiply by -1.

Finally, we need to multiply by \(-1\). $$-|19| = -19$$ The simplified expression is \(-19\).

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Key Concepts

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

Understanding Absolute Value
Absolute value is a fundamental concept in mathematics that refers to the non-negative value of a number without regard to its sign. It is often denoted by vertical bars, like \(|x|\). The absolute value of a number tells us how far it is from zero on the number line.
  • For example, \(|19| = 19\) because 19 is 19 units away from zero.
  • Similarly, \(|-19| = 19\) because -19 is also 19 units away from zero, albeit in the opposite direction.
The absolute value makes negative numbers positive because it measures "distance," which cannot be negative.
This is why when performing calculations within an absolute value, you need to ensure that once evaluated, the result inside becomes positive before any further operations.
Order of Operations: Multiplication and Addition
When simplifying expressions, it's crucial to follow the order of operations to ensure you reach the correct result. The standard sequence is PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). In this order:
  • Multiplication and division are performed before addition and subtraction.
  • For expressions on the same level, you solve them from left to right.
In our example, before adding \(-43\) and the result of \(6\cdot 4\), you first multiply: \(6\cdot 4=24\).
This results in the expression \(-43 + 24\) within the absolute value. Evaluating the multiplication separately makes it simpler and ensures accuracy.
The Simplification Process
Simplification in mathematics often involves reducing an expression to its simplest form. This is done by following clear steps based on arithmetic rules. In our given example, we simplify by following these steps:
  • Firstly, handle operations inside the absolute value by performing multiplication and then addition as per the order of operations.
  • Then, calculate the absolute value of the resultant number to get a non-negative result.
  • Finally, consider any remaining operations on the outside of the absolute value, such as multiplying by \(-1\) in this case to finish the simplification.
Proper simplification is essential for solving such problems correctly and can be achieved consistently by sticking to arithmetic rules and understanding how different mathematical concepts interact.

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Most popular questions from this chapter

For Exercises \(1-74,\) simplify. $$\frac{(5-19)+3^{3}}{(-2)(-6)-5^{2}}$$

Simplify. $$|12|$$

For Exercises \(1-74,\) simplify. $$\frac{3(-12)+1}{3^{2}-2}$$

For Exercises \(1-74,\) simplify. $$[19(-2)-(-18)] \div\\{15-5[2-(-1)]\\}$$

Simplify. $$|-47|$$
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