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Chapter 1: Problem 115

In the following exercises, simplify each expression. $$ 9-3|2(2-6)-(3-7)| $$

Short Answer

Expert verified
-3

Step by step solution

01

- Simplify inside the absolute value

First, simplify the expression inside the absolute value: \(2(2-6)-(3-7)\). Calculate the values inside the parentheses: \(2-6 = -4\) and \(3-7 = -4\). This gives us: \(2(-4)-(-4)\).
02

- Apply multiplication and simplify

Next, multiply \(2\) by \(-4\): \(2(-4)=-8\). Then, subtract \(-4\): \(-8 - (-4) = -8 + 4\). This results in \(-4\).
03

- Apply the absolute value function

Apply the absolute value function to \(-4\) to get \(|-4| = 4\).
04

- Simplify the entire expression

Finally, subtract \(3 \times 4\) from \(9\): \(9 - 3 \times 4 = 9 - 12 = -3\).

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Key Concepts

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

Absolute Value
Absolute value represents the distance of a number from zero, regardless of direction on the number line. It is always non-negative. To denote absolute value, we use vertical bars: \(|x|\). For example, the absolute value of -4 is \(|-4| = 4\).
When simplifying expressions, calculate the absolute value after performing operations inside the bars.
In the given exercise, we simplified the inside expression first: 2*(-4) - (-4) = -8 + 4 = -4. Then, apply the absolute value: \(|-4| = 4\).
Arithmetic Operations
Arithmetic operations involve addition, subtraction, multiplication, and division. These are the basic calculations we perform on numbers to get the desired result.
Let's look at the operations in our example. First, we performed multiplication: 2*(-4) = -8.
Then, addition or subtraction: -8 - (-4) = -8 + 4 = -4.
Simplification often requires a sequence of these operations.
Order of Operations
Order of operations is the set of rules that dictate which sequence steps should follow to solve an expression correctly. The common acronym for remembering this order is PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In our example, we start inside the parentheses: (2-6) and (3-7), then handle multiplication and absolute value next.
Parentheses
Parentheses are used in math to group parts of expressions and denote what should be calculated first, according to the order of operations. Calculations inside parentheses happen before any other operations outside them.
In the example, we simplified the expressions (2-6) and (3-7) first. This gave us -4 and -4, respectively. Then, these results were used for further calculations leading to the final simplification.

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