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Chapter 3: Problem 57

For the following exercises, perform the indicated operations. $$ [2+(-8)]-[5+(-7)] $$

Short Answer

Expert verified
Question: Evaluate and simplify the expression: $[2+(-8)]- [5+(-7)]$ Answer: The simplified expression is $$-4$$.

Step by step solution

01

Evaluating the expressions within the brackets

Evaluate the expressions within brackets first: $$[2+(-8)] =2-8$$ and $$[5+(-7)]=5-7$$.
02

Performing the operations within the brackets

Now, we will perform the subtraction within the brackets: $$2-8 = -6$$ and $$5-7 = -2$$.
03

Writing the new expression

Substitute the simplified expressions from step 2 into the original expression: $$ [-6]-[-2] $$
04

Subtracting the negative number

Recall that subtracting a negative number is the same as adding its positive counterpart. The expression becomes: $$-6+2 $$
05

Simplifying the final expression

Now, perform the addition: $$-6+2 = -4$$ The simplified expression is $$-4$$.

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Key Concepts

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

Integers
Integers are a set of numbers that include all the whole numbers, both positive and negative, as well as zero. This means they don't have fractional or decimal parts. When we think about integers, it's easy to envision a number line with zero in the middle, positive integers extending to the right, and negative integers extending to the left. Here are some common facts about integers:
  • Examples of integers include -3, 0, 4, and 7.
  • Integers are evenly spaced on a number line, making them easy to visualize.
  • Integers help represent many real-world quantities, like temperatures and elevations.
In algebra, integers are crucial because they are used to perform operations such as addition, subtraction, multiplication, and division. Understanding integers is fundamental for mastering more complex math topics.
Subtraction of Integers
Subtraction of integers might seem tricky because it involves both negative and positive numbers. It helps to think of subtraction as "adding the opposite." This means if you are subtracting a number, you can add its opposite instead. Let鈥檚 break down this important concept:
  • When subtracting a positive integer, like 3, from another number, you move left on the number line.
  • To subtract a negative integer, like -3, add the positive version of that number instead (so add 3), and move right on the number line.
  • For example, if you have the expression \(5 - (-3)\), rewrite it as \(5 + 3\), which equals 8.
This technique simplifies integer subtraction and makes it easier to handle complex calculations. By thinking of subtraction in this way, you can more confidently navigate through algebraic problems involving multiple integer operations.
Addition of Integers
Adding integers is usually straightforward, but it's essential to understand the effect of adding positive or negative numbers. Here are some key points to consider while adding integers:
  • When adding two positive integers, you simply combine their values, like \(3 + 5 = 8\).
  • When adding two negative integers, add their absolute values and make the result negative, such as \(-3 + (-2) = -5\).
  • When adding a positive integer and a negative integer, find the difference between their absolute values and take the sign of the number with the larger absolute value. For example, \(-6 + 2 = -4\).
These basic rules help break down more complicated expressions involving integers, ensuring you get the correct result each time. Recognizing how different signs interact is essential for solving a variety of algebraic equations and expressions.

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Most popular questions from this chapter

Perform the following operations. $$ \left(1.1 \times 10^{6}\right)^{2} $$

Perform the following operations. $$ \left(6 \times 10^{14}\right)\left(6 \times 10^{-10}\right) $$

Find the value of each of the following expressions. $$ -2[(4-8)-(5-11)] $$

For the following problems, convert the numbers from scientific notation to standard decimal form. There are about \(6 \times 10^{7}\) cells in the retina of the human eye.

Find the value of each of the following expressions. $$ -6+1-7 $$
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