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Chapter 1: Problem 14

Write each expression in words. a. \(-(-m)\) b. \(-2-(-3)\) c. \(x-(-y)\)

Short Answer

Expert verified
a. Opposite of the opposite of m b. Negative two plus three c. x plus y

Step by step solution

01

Understand the negative sign

In algebra, the negative sign in front of a variable or number indicates the opposite or the additive inverse. The expression \(-(-m)\) means the opposite of the opposite of \(m\). This is equivalent to \(m\) since the double negative cancels out.
02

Simplify the expression with negative numbers

The expression \(-2-(-3)\) involves subtracting a negative number. Subtracting a negative number is equivalent to adding the positive equivalent of that number. Thus, \(-2 - (-3)\) simplifies to \(-2 + 3\), which equals 1.
03

Address the subtraction of negative variables

For the expression \(x-(-y)\), subtracting the negative \(y\) is the same as adding \(y\) to \(x\) because subtracting a negative is equivalent to addition. This expression simplifies to \(x + y\).

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Key Concepts

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

Algebraic Expressions
In math, an algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, and multiplication. These expressions are the building blocks of algebra.
  • Variables are symbols that stand for unknown values. Usually denoted by letters like x, y, or m.
  • Numbers in the expressions can be positive or negative.
  • Operations connect the numbers and variables. For example, "x + 3" is a simple algebraic expression.
Expressions can be simplified by using algebraic rules and principles, which helps in solving algebra problems or translating expressions to words. For instance, the expression x-(-y) can be read as "x minus negative y", which simplifies to "x plus y" due to the nature of subtraction and negative numbers.
Additive Inverse
The additive inverse of a number is crucial in algebra because it helps us understand negative signs. The additive inverse of a number is what you add to it to get zero.
  • For any number m, its additive inverse is -m. Together they add up to zero: \( m + (-m) = 0 \).
  • The opposite of -m is m, which is just another way of saying m is the additive inverse of -m.
  • Negative signs in expressions indicate using the additive inverse. For example, -(-m) translates to the additive inverse of -m, which leads back to m.
Understanding additive inverses is key to simplifying expressions and solving equations, allowing us to translate and rework expressions like \(-(-m)\) back into simpler terms.
Double Negative
The concept of a double negative is straightforward. In math, a double negative cancels out to a positive, just like in language.
  • If you see -(-x), it means taking the opposite of the opposite of x, which is just x.
  • Statements like -2-(-3) involve changing subtraction of a negative to addition: -2 + 3. This simplifies because subtracting a negative is equivalent to adding the positive.
  • In verbal expressions, these simplify to "the opposite of the opposite." This is useful when interpreting or simplifying algebraic expressions, keeping in mind that two negatives make a positive.
Remembering that double negatives turn into positives helps us interpret and simplify expressions, like transforming \(-(-m)\) and \(x-(-y)\) into clearer expressions for manageable calculations.

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Is the following statement true or false? Explain. Having a debt of \(\$ 100\) forgiven is equivalent to gaining \(\$ 100\).

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True or false: \(-4>-5 ?\)
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