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Chapter 1: Problem 18

Find each sum. $$ 3+(-7) $$

Short Answer

Expert verified
The sum of 3 and -7 is -4.

Step by step solution

01

Identify the Numbers

Observe the two numbers that need to be added: 3 and -7.
02

Understand the Operation

Since one number is positive and the other is negative, this operation will essentially be a subtraction.
03

Subtract the Numbers

Subtract the smaller absolute value from the larger absolute value: 7 - 3 = 4. The absolute values are taken for subtraction.
04

Determine the Sign

Since -7 has a larger absolute value than 3, the result will carry the sign of -7. Therefore, the result is -4.

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on a number line, regardless of direction. This means it鈥檚 always non-negative. For instance, the absolute value of 3 is written as \(|3| = 3\) and for -7, it鈥檚 written as \(|-7| = 7\).

Absolute values are used to simplify operations involving positive and negative numbers by focusing on their magnitude.
So when solving \(3 + (-7)\), the absolute values of 3 and -7 help us determine the numerical steps without initially considering their signs.
Positive and Negative Numbers
Positive numbers are greater than zero, and negative numbers are less than zero.
For the problem \(3 + (-7)\), 3 is positive and -7 is negative.
When combining positive and negative numbers, think of it as moving in different directions on a number line: adding a negative is like moving left, while adding a positive is like moving right.

This concept turns the addition \(3 + (-7)\) into a form where we understand that the operation essentially becomes a subtraction due to the different signs:
  • Addition of positive and negative numbers cancels out a portion of each value.
  • The sign of the larger absolute value (here, -7) determines the sign of the result.
Subtraction Operation
Subtraction involves finding the difference between two numbers. When adding a positive and a negative number, similar to \(3 + (-7)\), you subtract the smaller absolute value from the larger absolute value.
  • First, find the absolute values: \(3\) and \(7\).
  • Then subtract: \(7 - 3 = 4\).

Since -7 is the larger number in magnitude and it鈥檚 negative, the result carries this sign, resulting in the final answer of \(-4\). This step-by-step process makes it easier to handle operations involving different signs.

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Most popular questions from this chapter

A manufacturer has fixed costs of \(\$ 1000\) to produce gizmos. Each gizmo costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the cost to make \(y\) gadgets.

Use the distributive property to rewrite each expression. $$ -\frac{1}{3}(9 x-4) $$

Use the distributive property to rewrite each expression. $$ 7(z-8) $$

Solve each problem. The average annual spending per U.S. household on apparel and apparel services was \(\$ 1786\) in 2014. This amount increased \(\$ 60\) by 2015 and then decreased \(\$ 43\) by 2016 What was the average household expenditure for apparel and apparel services in \(2016 ?\) (Data from U.S. Bureau of Labor Statistics.)

Solve each problem. In 2005, Americans saved \(-0.5 \%\) of their after-tax incomes. In September 2017 , they saved \(3.1 \%\). (Data from U.S. Bureau of Economic Analysis.) (a) Express the difference between these amounts as a positive number. (b) How could Americans have a negative personal savings rate in \(2005 ?\)
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