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Chapter 1: Problem 36

Perform the following operations with real numbers. $$-32.6-(-9.8)$$

Short Answer

Expert verified
-22.8

Step by step solution

01

Identify the Operation and the Numbers

The given expression is \(-32.6 - (-9.8)\). This involves subtracting a negative number from a negative number.
02

Simplify the Double Negative

When you subtract a negative number, it is equivalent to adding the positive form of that number. Convert the expression \(-32.6 - (-9.8)\) to \(-32.6 + 9.8\).
03

Perform the Addition

Now, perform the operation \(-32.6 + 9.8\). When you add a positive number (9.8) to a negative number (-32.6), the result will be negative but closer to zero. Calculate: \(-32.6 + 9.8 = -22.8\).

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Key Concepts

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

Subtracting Negative Numbers
When we come across the operation of subtracting a negative number, it can be a bit counterintuitive at first. Think of negative numbers as having direction, much like temperature below zero or an elevator moving downwards. The negative sign indicates a move in the opposite direction of the positive numbers. So, when you subtract a negative, you are essentially removing the effect of that negative value.
  • Subtracting \(-(-9.8)\) is like saying "Let's move in the positive direction by 9.8."
  • This changes the operation from subtraction to addition.
Understanding this can clear up a lot of confusion, especially when solving problems involving multiple operations with real numbers.
Addition with Negative Numbers
Adding a negative number is another concept that can be a little tricky. When you add a negative number, you are effectively decreasing the total, moving the value further away from zeros on the number line. Consider an elevator analogy: if you're at the 5th floor and you go down 2 floors, you end up on the 3rd floor.
  • Imagine \(-32.6 + 9.8\), where you add a positive to a negative.
  • The positive number acts as a counterforce, reducing the overall negative impact.
  • Think of it like summing debts and credits: each positive (credit) offsets part of the negative (debt).
This step-by-step approach to adding negative and positive numbers will build a better understanding of how real-number operations work.
Double Negative Simplification
Simplifying double negatives is an essential part of mastering operations with real numbers. When you encounter two negative signs in a mathematical expression, it's important to remember that they transform into a positive. Just as in language where a double negative implies a positive, the same applies in math.
  • For example, \(-(-9.8)\) equals \(+9.8\).
  • A practical approach is to cancel out the negatives, making it simpler to perform arithmetic operations.
  • This concept is very useful in algebra, calculus, and even in day-to-day problem-solving.
By mastering the technique of turning these double negatives into positives, solving expressions becomes straightforward and less error-prone.

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Most popular questions from this chapter

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(7 a-2 b-9 a+3 b, \quad a=4\) and \(b=-6\)

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$5(2 x+1)+4(3 x-2)$$

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(2(a+b)^{2}, \quad a=6\) and \(b=-1\)

Use your calculator and evaluate each of the algebraic expressions for the indicated values. Express the final answers to the nearest tenth. \(\pi r^{2} h, \quad \pi=3.14, r=1.6\), and \(h=11.2\)

Use your calculator and evaluate each of the algebraic expressions for the indicated values. Express the final answers to the nearest tenth. \(\pi r^{2} h, \quad \pi=3.14, r=4.8\), and \(h=15.1\)
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