Adding the additive inverse instead of the opposite

Oct 13, 2025 by miodeya.com 109 views

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires careful attention to detail. In this article, we will explore a common mistake made by students when simplifying expressions and provide a step-by-step guide on how to correct it.

The Student's Error

A student was given the expression $(6x - 2) - (-x + 5)$ and simplified it to $7x + 3$ . However, the student made an error in the simplification process. To find the error, let's break down the expression and simplify it step by step.

Step 1: Distribute the Negative Sign

When subtracting a negative term, we can distribute the negative sign to the terms inside the parentheses. In this case, we have:

(6x - 2) - (-x + 5) = 6x - 2 + x - 5

Step 2: Combine Like Terms

Now, we can combine like terms by adding or subtracting the coefficients of the same variables. In this case, we have:

6x + x - 2 - 5 = 7x - 7

Step 3: Identify the Error

Comparing the student's answer, $7x + 3$ , with our correct simplification, $7x - 7$ , we can see that the student made an error. The student added the additive inverse of $-2$ and $-5$ instead of adding their opposites.

Correct Simplification

To correct the student's error, we need to add the additive inverse of $-2$ and $-5$ , which are $2$ and $5$ , respectively. The correct simplification is:

(6x - 2) - (-x + 5) = 7x - 7

Conclusion

In conclusion, simplifying algebraic expressions requires careful attention to detail and a thorough understanding of the rules of arithmetic. By following the steps outlined in this article, students can avoid common mistakes and arrive at the correct solution.

Common Mistakes to Avoid

When simplifying algebraic expressions, students often make the following mistakes:

Adding the additive inverse instead of the opposite: This is the mistake made by the student in this example. To avoid this mistake, students need to remember that the additive inverse of a term is the term that, when added to it, results in zero.
Not distributing the negative sign: When subtracting a negative term, students need to distribute the negative sign to the terms inside the parentheses.
Not combining like terms: Students need to combine like terms by adding or subtracting the coefficients of the same variables.

Tips for Simplifying Algebraic Expressions

To simplify algebraic expressions, students can follow these tips:

Read the expression carefully: Before simplifying the expression, students need to read it carefully and understand what is being asked.
Distribute the negative sign: When subtracting a negative term, students need to distribute the negative sign to the terms inside the parentheses.
Combine like terms: Students need to combine like terms by adding or subtracting the coefficients of the same variables.
Check the answer: Finally, students need to check their answer to make sure it is correct.

Practice Problems

To practice simplifying algebraic expressions, students can try the following problems:

Simplify the expression $(3x + 2) - (2x - 3)$ .
Simplify the expression $(x - 2) - (x + 4)$ .
Simplify the expression $(2x + 5) - (x - 2)$ .

Solutions

Here are the solutions to the practice problems:

$(3x + 2) - (2x - 3) = 3x + 2 - 2x + 3 = x + 5$
$(x - 2) - (x + 4) = x - 2 - x - 4 = -6$
$(2x + 5) - (x - 2) = 2x + 5 - x + 2 = x + 7$

Conclusion

In conclusion, simplifying algebraic expressions requires careful attention to detail and a thorough understanding of the rules of arithmetic. By following the steps outlined in this article and practicing with the provided problems, students can avoid common mistakes and arrive at the correct solution.

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires careful attention to detail. In this article, we will answer some frequently asked questions (FAQs) on simplifying algebraic expressions.

Q: What is the difference between adding and subtracting algebraic expressions?

A: When adding algebraic expressions, we combine like terms by adding or subtracting the coefficients of the same variables. When subtracting algebraic expressions, we distribute the negative sign to the terms inside the parentheses and then combine like terms.

Q: How do I simplify an expression with multiple negative signs?

A: When simplifying an expression with multiple negative signs, we need to distribute the negative signs to the terms inside the parentheses and then combine like terms. For example, $(3x - 2) - (-x + 5)$ can be simplified as follows:

(3x - 2) - (-x + 5) = 3x - 2 + x - 5 = 4x - 7

Q: What is the difference between the additive inverse and the opposite?

A: The additive inverse of a term is the term that, when added to it, results in zero. For example, the additive inverse of $3x$ is $-3x$ . The opposite of a term is the term that, when multiplied by it, results in a positive number. For example, the opposite of $-3x$ is $-3x$ .

Q: How do I simplify an expression with parentheses?

A: When simplifying an expression with parentheses, we need to follow the order of operations (PEMDAS):

Parentheses: Evaluate the expressions inside the parentheses first.
Exponents: Evaluate any exponents next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example, $x$ is a variable. A constant is a value that does not change. For example, $5$ is a constant.

Q: How do I simplify an expression with variables and constants?

A: When simplifying an expression with variables and constants, we need to combine like terms by adding or subtracting the coefficients of the same variables. For example, $3x + 5 - 2x$ can be simplified as follows:

3x + 5 - 2x = x + 5

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression with one variable and a degree of one. For example, $3x + 2$ is a linear expression. A quadratic expression is an expression with one variable and a degree of two. For example, $x^2 + 3x + 2$ is a quadratic expression.

Q: How do I simplify an expression with a variable raised to a power?

A: When simplifying an expression with a variable raised to a power, we need to follow the rules of exponents. For example, $(x^2 + 3x + 2) - (x + 1)$ can be simplified as follows:

(x^2 + 3x + 2) - (x + 1) = x^2 + 3x + 2 - x - 1 = x^2 + 2x + 1

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two polynomials. For example, $\frac{x + 2}{x - 1}$ is a rational expression. An irrational expression is an expression that cannot be written as a fraction of two polynomials. For example, $\sqrt{x + 2}$ is an irrational expression.

Q: How do I simplify an expression with a rational or irrational term?

A: When simplifying an expression with a rational or irrational term, we need to follow the rules of algebra. For example, $\frac{x + 2}{x - 1} + \sqrt{x + 2}$ can be simplified as follows:

\frac{x + 2}{x - 1} + \sqrt{x + 2} = \frac{x + 2}{x - 1} + \frac{\sqrt{x + 2}}{1}

Conclusion

In conclusion, simplifying algebraic expressions requires careful attention to detail and a thorough understanding of the rules of algebra. By following the steps outlined in this article and practicing with the provided problems, students can avoid common mistakes and arrive at the correct solution.