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Chapter 1: Problem 45

Add or subtract as indicated. $$|7-11|+|6+(-13)|$$

Short Answer

Expert verified
The short answer for the given question is: Simplify the expressions inside the absolute value functions to get \(|-4|\) and \(|-7|\). Calculate the absolute values which are \(4\) and \(7\). Finally, add the absolute values to get the final result, \(11\).

Step by step solution

01

Simplify the expressions inside the absolute value functions

Evaluate the expressions inside the absolute value bars: \( |7-11| \) and \( |6+(-13)| \)
02

Calculate the absolute values

Find the absolute values of the simplified expressions: \( |-4| \) and \( |-7| \)
03

Add the absolute values

Now, add the absolute values and find the final result: \( 4+7 \)
04

Final result

The final result after adding the absolute values is: \( 11 \)

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Key Concepts

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

Integer Operations
Integer operations involve the arithmetic processes we use with whole numbers, such as addition and subtraction. Integers include all positive numbers, negative numbers, and zero. When combining integers, it's crucial to understand:
  • Sign Rules: These dictate how the signs of the integers affect the operation. Adding two positive integers or two negative integers will yield a positive or negative result, respectively. But adding a positive and a negative integer depends on the magnitude of each number.
  • Interaction with Absolute Value: Absolute value tells us the distance of a number from zero on the number line, without considering direction. Thus, operations with integers often involve absolute values to ensure results are understood in terms of magnitude irrespective of sign.
Understanding these rules helps simplify complex expressions by focusing on magnitude first, followed by evaluating the overall sign for clearer results.
Mathematical Expressions
A mathematical expression is a combination of numbers, symbols, and operators that indicate a calculation to be performed. In the example, we see the expression \( |7-11|+|6+(-13)| \), which involves absolute values and the operations of addition and subtraction.

  • Parentheses and Absolute Value Bars: Both of these act as grouping symbols, showing which operations should be performed first according to the order of operations.
  • Order of Operations: It's necessary to address expressions inside parentheses or absolute value bars before performing other operations, ensuring accurate evaluation.
Expressions form the basis for creating equations and solving mathematical problems by setting a clear sequence for performing calculations.
Addition and Subtraction of Numbers
Addition and subtraction are fundamental mathematical operations. They are inverse operations: addition combines values into a larger quantity, while subtraction finds the difference between values.

  • Addition Process: For addition, especially when involving absolute values, we consider the sum of the magnitudes of numbers as seen in the step of adding 4 and 7 to get 11.
  • Subtraction Process: Subtraction reverses addition, and when integers are involved, handling negative signs is critical to simplify computations, such as turning \( 6 + (-13) \) into a clear operation to evaluate as \(-7\).
  • Connection to Absolute Values: When dealing with absolute values, you first simplify numbers inside the bars, convert them into positive magnitudes, then perform addition or subtraction on these values.
Mastering these operations alongside understanding absolute values ensures efficient and accurate solving of diverse mathematical problems.

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Most popular questions from this chapter

Use a signed number to represent the change in each situation. In 2004 , the median income for females 25 years and older with a bachelor's degree was \(\$ 31,585,\) and in 2005 it increased by \(\$ 1083\) to \(\$ 32,668\). (U.S. Census Bureau)

Rewrite each expression using the distributive property. Simplify if possible. $$-7(2-6)$$

Rewrite each expression using the distributive property. Simplify if possible. $$10(m+5 n-3)$$

Write an expression for each and simplify. Answer the question with a complete sentence. In \(2002,\) the total attendance at Major League Baseball games was \(67,859,176 .\) In \(2003,\) that figure was \(67,630,052 .\) What was the difference in the number of people who went to ballparks from 2002 to \(2003 ?\)

Use a signed number to represent the change in each situation. In Michael Jordan's last season with the Chicago Bulls \((1997-1998)\) the average attendance at the United Center was \(23,988 .\) During the \(2002-2003\) season, the average attendance fell to \(19,617 .\) This was a decrease of 4371 people per game. (Total Basketball: The Ultimate Basketball Encyclopedia, by Ken Shouler, Bob Ryan, Sam Smith, Leonard Koppett, and Bob Bellotti, \(\oplus\) 2003, SPORT Media Pub, Inc.)
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