Problem 4
Question
Consider the subtraction \(2-(-8)\) a. Find the opposite, or additive inverse, of \(-8\) b. Rewrite the subtraction as the addition of the opposite of \(-8\)
Step-by-Step Solution
Verified Answer
For part a, the opposite of \(-8\) is \(8\). For part b, the subtraction \(2-(-8)\) can be rewritten as the addition \(2 + 8\).
1Step 1: Identify the opposite of -8
The opposite or additive inverse of a number is the number that, when added to the original number, equals zero. Hence, the opposite of \(-8\) would be \(8\), because \(-8 + 8 = 0\).
2Step 2: Rewrite the subtraction as an addition
The subtraction \(2-(-8)\) can be rewritten as the addition of the opposite of \(-8\). So, \(2-(-8)\) can be rewritten as \(2 + 8\).
Key Concepts
Subtraction and AdditionOpposite of a NumberAlgebraic Manipulation
Subtraction and Addition
Understanding subtraction and addition is fundamental in mathematics. Subtraction, which is often seen as the process of taking away, can also be approached as adding the opposite. This means that when facing a subtraction like \(2 - (-8)\), you can think of it as adding the opposite of the second number. In this case, the opposite of \(-8\) is \(8\), because they are equidistant from zero on the number line but in different directions.
When you're asked to switch from subtraction to addition, as in the exercise, you're not changing the value of the expression, just the way you look at it. Instead of taking away a negative number, you're adding its positive counterpart. This simplifies calculations and helps avoid common mistakes when dealing with negative numbers. Visualizing a number line can be a helpful tool, as it allows you to see the movement from one number to the next, clarifying the subtractive or additive relationship between them.
When you're asked to switch from subtraction to addition, as in the exercise, you're not changing the value of the expression, just the way you look at it. Instead of taking away a negative number, you're adding its positive counterpart. This simplifies calculations and helps avoid common mistakes when dealing with negative numbers. Visualizing a number line can be a helpful tool, as it allows you to see the movement from one number to the next, clarifying the subtractive or additive relationship between them.
Opposite of a Number
The concept of the 'opposite of a number', or its additive inverse, is crucial for understanding subtraction and addition in algebra. The opposite of any number \(x\) is the number that, when added to \(x\), will result in zero. It's like finding a counterpart for every number on a number line that balances it out perfectly. For example, if you have the number \(3\), its opposite would be \(-3\) because \(3 + (-3) = 0\).
This concept isn't limited to positive numbers; it applies equally to negatives, as seen in the exercise with the number \(-8\). The opposite of \(-8\) is \(8\), because their sum is zero. By understanding this concept, you can convert tricky subtraction problems into straightforward addition problems, which can be easier to solve and understand.
This concept isn't limited to positive numbers; it applies equally to negatives, as seen in the exercise with the number \(-8\). The opposite of \(-8\) is \(8\), because their sum is zero. By understanding this concept, you can convert tricky subtraction problems into straightforward addition problems, which can be easier to solve and understand.
Algebraic Manipulation
Algebraic manipulation involves rewriting expressions in a form that makes them easier to work with or understand. This can include factoring, distributing, combining like terms, or, as in the given exercise, turning subtraction into addition. The ultimate goal of these manipulations is to simplify the expression or to solve for an unknown variable.
Being adept at algebraic manipulation requires an understanding of the underlying properties of numbers, like the distributive property, the commutative property, and the concept of additive inverses. Knowing that you can express subtraction as the addition of an opposite allows you to transform expressions, thus simplifying complex algebraic problems into more manageable equations. This skill is not only useful in solving equations but also in various applications of mathematics, from calculating finances to understanding scientific formulas.
Being adept at algebraic manipulation requires an understanding of the underlying properties of numbers, like the distributive property, the commutative property, and the concept of additive inverses. Knowing that you can express subtraction as the addition of an opposite allows you to transform expressions, thus simplifying complex algebraic problems into more manageable equations. This skill is not only useful in solving equations but also in various applications of mathematics, from calculating finances to understanding scientific formulas.
Other exercises in this chapter
Problem 3
Evaluate each expression for \(x=4\). $$12-x$$
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Convert each mixed number to an improper fraction. $$7 \frac{3}{5}$$
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In Exercises \(1-14\), evaluate each exponential expression. $$6^{3}$$
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An algebraic expression is given. Use each expression to answer the following questions. a. How many terms are there in the algebraic expression? b. What is the
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