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Chapter 1: Problem 6

Evaluate each of the given expressions by performing the indicated operations. $$-4+(-7)$$

Short Answer

Expert verified
The expression evaluates to \(-11\).

Step by step solution

01

Identify the Operation

In the expression \(-4 + (-7)\), the operation between the two numbers is addition. However, since both numbers are negative, this is equivalent to adding two negative numbers.
02

Combine the Numbers

Both numbers are negative, so when you add them, you simply add their absolute values. Thus, the absolute value of \(-4\) is \(4\), and the absolute value of \(-7\) is \(7\). Adding them gives \(4 + 7 = 11\).
03

Apply the Negative Sign

Since we are adding two negative numbers, the sum is also negative. Therefore, \(-4 + (-7) = -11\).

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Key Concepts

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

Negative Numbers
Negative numbers are essential in mathematics because they represent values less than zero. They are written with a minus sign in front of them, like
  • -4
  • -7
These numbers are not just theoretical. They have practical uses as well, such as denoting temperatures below freezing or negative balances in an account. Another important aspect to remember is:
  • A negative number is the opposite of its positive counterpart. For example, \( -4 \) is the opposite of \( 4 \).
Understanding how negative numbers behave is crucial for solving equations that involve them. For instance, when you add two negative numbers, you move further away from zero, deepening into the negative side of the number line.
Addition of Integers
The process of adding integers involves combining whole numbers, both positive and negative. In the expression \(-4 + (-7)\), we are adding two negative integers. The steps for adding generally are:
  • Determine the signs of the integers involved. If both are negative, their sum increases the negative value.
  • Find the absolute value of each integer. This is simply the number without its negative sign.
  • Add the absolute values together.
  • Apply the negative sign to the sum if both integers were negative, resulting in a negative final answer.
In our example, these steps result in \(-11\), highlighting that two negatives combined breed a more profound negative value.
Absolute Value
The absolute value of a number is its distance from zero on the number line. It disregards any negative sign and treats every number as non-negative. For instance:
  • The absolute value of \(-4\) is \(4\)
  • The absolute value of \(-7\) is \(7\)
This concept is crucial when dealing with negative numbers in arithmetic operations. When tasked with adding negative integers, such as in the expression \(-4 + (-7)\), you first compute their absolute values. This simplifies your calculation to \(4 + 7\). Ultimately, once you know the absolute value sum, you can easily adjust the final sign depending on the original numbers' signs, allowing for a clear path to the correct answer even as the numbers themselves shift on the number line.

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