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Chapter 3: Problem 61

For the following exercises, perform the indicated operations. $$ (5-12)-(4-10) $$

Short Answer

Expert verified
Answer: -1

Step by step solution

01

Simplify inside the brackets

Start by simplifying the expressions inside the brackets. Calculate the results of each subtraction inside the brackets. $$ (5-12)=-7 $$ and $$ (4-10)=-6 $$
02

Perform the subtraction

Now that we have simplified the expressions inside the brackets, we can perform the subtraction. $$ (-7) - (-6) $$
03

Calculate the result

Remember that subtracting a negative number is the same as adding its positive counterpart. Therefore, our expression becomes $$ -7 + 6 $$ The result is $$ -1 $$ The final answer is -1.

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Key Concepts

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

Subtraction
Subtraction is one of the most basic arithmetic operations. It involves finding the difference between two numbers, which means restarting or reducing the value of another number. In our exercise, we start by subtracting two pairs of numbers that are inside the brackets:
- The first bracket is \((5 - 12)\), and the second is \((4 - 10)\).
  • In each bracket, the number being subtracted is larger than the number it's subtracted from, leading to a negative result.
  • Subtraction is often used in daily tasks, such as calculating change or determining how much more time one needs to wait.
Negative Numbers
Negative numbers are simply numbers with a value less than zero. They are often represented with a minus sign (e.g., -1, -7, -6).
In our exercise, when we subtract larger numbers from smaller ones in each bracket, we end up with negative results:
  • \((5 - 12) = -7\)
  • \((4 - 10) = -6\)
Negative numbers are useful in various scenarios, such as measuring temperature below freezing or financial losses.
When dealing with negative numbers in operations, it is crucial to pay attention to the signs to ensure accurate calculations.
Simplification
Simplification involves making mathematical expressions easier to understand and solve. In our exercise, we first simplify the expressions within the brackets:
  • From \((5 - 12)\), we simplify it to \(-7\).
  • From \((4 - 10)\), we simplify it to \(-6\).
Next, these simplified results help us perform further operations more easily. Simplification not only clears away complexity but also reduces the chance of calculation errors. It allows us to see the core operations needed to solve an equation.
Elementary Algebra
Elementary algebra deals with basic algebraic concepts, such as variables, constants, and simple operations like addition, subtraction, multiplication, and division. In our exercise, we explore subtraction with negative numbers using basic algebraic rules:
  • Once simplified, we express the problem as \((-7) - (-6)\).
  • We use the concept that subtracting a negative is equivalent to adding its positive: hence \(-7 + 6\).
  • This final operation gives us a result of \(-1\).
This exercise illustrates how crucial understanding each algebraic operation concept is to solving mathematical problems effectively. Mastering these basics prepares you for more complex algebraic equations and applications.

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