Which Expression Is Equivalent To $6(x-4)$?A. $-6x + 4$ B. $ 6 X − 4 6x - 4 6 X − 4 [/tex] C. $6x - 24$ D. $-6x + 24$
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression $6(x-4)$. We will examine the different options provided and determine which one is equivalent to the given expression.
Understanding the Given Expression
The given expression is $6(x-4)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b+c) = ab + ac.
Applying the Distributive Property
Using the distributive property, we can rewrite the given expression as:
Simplifying the Expression
Now, we can simplify the expression by evaluating the product of 6 and 4:
Therefore, the simplified expression is:
Evaluating the Options
Now that we have simplified the expression, let's evaluate the options provided:
- A. $-6x + 4$
- B. $6x - 4$
- C. $6x - 24$
- D. $-6x + 24$
Option A: $-6x + 4$
Option A is incorrect because it has a negative sign in front of the x term, whereas our simplified expression has a positive sign.
Option B: $6x - 4$
Option B is incorrect because it is missing the constant term -24, which is present in our simplified expression.
Option C: $6x - 24$
Option C is correct because it matches our simplified expression exactly.
Option D: $-6x + 24$
Option D is incorrect because it has a negative sign in front of the x term, whereas our simplified expression has a positive sign.
Conclusion
In conclusion, the correct answer is option C: $6x - 24$. This expression is equivalent to the given expression $6(x-4)$. We applied the distributive property and simplified the expression to arrive at this result.
Tips and Tricks
- When simplifying algebraic expressions, always apply the distributive property to remove parentheses.
- Be careful with negative signs and make sure to distribute them correctly.
- Simplify expressions by combining like terms and evaluating products.
Practice Problems
Try simplifying the following expressions:
Answer Key
Final Thoughts
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, a(b+c) = ab + ac. This means that we can distribute a single term to multiple terms inside parentheses.
Q: How do I apply the distributive property?
A: To apply the distributive property, simply multiply the single term to each term inside the parentheses. For example, if we have the expression 3(2x+1), we would multiply 3 to each term inside the parentheses: 3(2x) + 3(1).
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, 4 is a constant.
Q: How do I simplify an expression with variables and constants?
A: To simplify an expression with variables and constants, we need to apply the distributive property and combine like terms. Like terms are terms that have the same variable and exponent. For example, 2x and 3x are like terms.
Q: What is the order of operations?
A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions 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: How do I evaluate an expression with exponents?
A: To evaluate an expression with exponents, we need to follow the order of operations. For example, if we have the expression 2^3, we would evaluate the exponent first: 2^3 = 8.
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 1. For example, 2x is a linear expression. A quadratic expression is an expression with one variable and a degree of 2. For example, x^2 + 3x is a quadratic expression.
Q: How do I simplify a quadratic expression?
A: To simplify a quadratic expression, we need to apply the distributive property and combine like terms. We can also use factoring to simplify a quadratic expression.
Q: What is factoring?
A: Factoring is a process of expressing an expression as a product of simpler expressions. For example, if we have the expression x^2 + 4x, we can factor it as x(x+4).
Q: How do I factor a quadratic expression?
A: To factor a quadratic expression, we need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, if we have the expression x^2 + 4x, we can factor it as x(x+4).
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, x/2 is a rational expression. An irrational expression is an expression that cannot be written as a fraction of two polynomials. For example, √2 is an irrational expression.
Q: How do I simplify a rational expression?
A: To simplify a rational expression, we need to apply the distributive property and combine like terms. We can also use factoring to simplify a rational expression.
Q: What is the difference between a polynomial and a non-polynomial expression?
A: A polynomial expression is an expression that can be written as a sum of terms, where each term is a product of a variable and a constant. For example, x^2 + 3x is a polynomial expression. A non-polynomial expression is an expression that cannot be written as a sum of terms, where each term is a product of a variable and a constant. For example, sin(x) is a non-polynomial expression.
Q: How do I simplify a non-polynomial expression?
A: To simplify a non-polynomial expression, we need to use trigonometric identities or other mathematical techniques to rewrite the expression in a simpler form.
Conclusion
In conclusion, simplifying algebraic expressions is an essential skill in mathematics. By applying the distributive property, combining like terms, and using factoring, we can simplify expressions and arrive at equivalent expressions that are easier to work with. Remember to be careful with negative signs and to simplify expressions by combining like terms and evaluating products. With practice, you will become proficient in simplifying algebraic expressions and be able to tackle more complex problems with confidence.