Which Expressions Are Equivalent To $-9\left(\frac{2}{3} X+1\right$\]? Check All That Apply.A. $-9\left(\frac{2}{3} X\right) + 9(1$\]B. $-9\left(\frac{2}{3} X\right) - 9(1$\]C. $-9\left(\frac{2}{3} X\right) + 1$D.

Understanding the Problem

When simplifying algebraic expressions, it's essential to understand the rules of distribution and the properties of exponents. In this article, we'll explore the concept of equivalent expressions and how to simplify complex algebraic expressions.

What are Equivalent Expressions?

Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms. In other words, equivalent expressions are expressions that can be transformed into each other through a series of algebraic operations.

The Given Expression

The given expression is $-9\left(\frac{2}{3} x+1\right)$. 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:

$$-9\left(\frac{2}{3} x+1\right) = -9\left(\frac{2}{3} x\right) + (-9)(1)$$

Simplifying the Expression

Now, let's simplify the expression further by applying the rules of arithmetic operations.

$$-9\left(\frac{2}{3} x\right) + (-9)(1) = -6x - 9$$

Checking the Options

Now that we have simplified the expression, let's check the options to see which ones are equivalent to the given expression.

Option A: $-9\left(\frac{2}{3} x\right) + 9(1)$

This option is not equivalent to the given expression because it lacks the negative sign in front of the second term.

Option B: $-9\left(\frac{2}{3} x\right) - 9(1)$

This option is equivalent to the given expression because it has the same value as the simplified expression.

Option C: $-9\left(\frac{2}{3} x\right) + 1$

This option is not equivalent to the given expression because it lacks the negative sign in front of the second term.

Option D: $-6x - 9$

This option is equivalent to the given expression because it has the same value as the simplified expression.

Conclusion

In conclusion, the correct options that are equivalent to the given expression are B and D. These options have the same value as the simplified expression, while options A and C do not.

Key Takeaways

• Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms.
• The distributive property is a fundamental concept in algebra that allows us to simplify complex expressions.
• When simplifying expressions, it's essential to apply the rules of arithmetic operations and to check the options to see which ones are equivalent to the given expression.

Frequently Asked Questions

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 $c$, $a(b+c) = ab + ac$.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you need to apply the distributive property and the rules of arithmetic operations.

Q: What are equivalent expressions?

A: Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms.

Q: How do I check if an expression is equivalent to the given expression?

A: To check if an expression is equivalent to the given expression, you need to simplify the expression and compare it to the given expression.

Additional Resources

• Khan Academy: Algebra
• Mathway: Algebra
• Wolfram Alpha: Algebra

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Linear Algebra" by Jim Hefferon
  Frequently Asked Questions: Algebraic Expressions =====================================================

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 property allows us to simplify complex expressions by distributing the terms inside the parentheses.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you need to apply the distributive property and the rules of arithmetic operations. Here are the steps to follow:

1. Distribute the terms inside the parentheses using the distributive property.
2. Combine like terms by adding or subtracting the coefficients of the same variables.
3. Simplify the expression by applying the rules of arithmetic operations.

Q: What are equivalent expressions?

A: Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms. For example, the expressions $2x + 3$ and $x + 2x + 3$ are equivalent because they have the same value.

Q: How do I check if an expression is equivalent to the given expression?

A: To check if an expression is equivalent to the given expression, you need to simplify the expression and compare it to the given expression. Here are the steps to follow:

1. Simplify the expression by applying the distributive property and the rules of arithmetic operations.
2. Compare the simplified expression to the given expression.
3. If the expressions have the same value, then they are equivalent.

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

A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same. For example, in the expression $2x + 3$, $x$ is a variable and $3$ is a constant.

Q: How do I evaluate an expression?

A: To evaluate an expression, you need to substitute the values of the variables into the expression and then simplify it. Here are the steps to follow:

1. Substitute the values of the variables into the expression.
2. Simplify the expression by applying the rules of arithmetic operations.
3. Evaluate the expression to find the final value.

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:

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

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to follow these steps:

1. Simplify the fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
2. Combine like terms by adding or subtracting the coefficients of the same variables.
3. Simplify the expression by applying the rules of arithmetic operations.

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

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation $2x + 3 = 5$ is a linear equation, while the equation $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by applying the rules of arithmetic operations. Here are the steps to follow:

1. Add or subtract the same value to both sides of the equation to isolate the variable.
2. Multiply or divide both sides of the equation by the same value to isolate the variable.
3. Simplify the equation to find the final value of the variable.

Q: What is the difference between a function and a relation?

A: A function is a relation in which each input value corresponds to exactly one output value, while a relation is a set of ordered pairs that may have multiple input values corresponding to the same output value. For example, the relation $\{(2,3),(4,5),(6,7)\}$ is a relation, while the function $f(x) = 2x + 1$ is a function.

Q: How do I graph a function?

A: To graph a function, you need to follow these steps:

1. Identify the type of function, such as linear, quadratic, or polynomial.
2. Determine the domain and range of the function.
3. Plot the points on a coordinate plane.
4. Draw a smooth curve through the points to represent the function.

Q: What is the difference between a dependent variable and an independent variable?

A: A dependent variable is a variable that depends on the value of another variable, while an independent variable is a variable that is not dependent on the value of another variable. For example, in the equation $y = 2x + 1$, $y$ is the dependent variable and $x$ is the independent variable.

Q: How do I determine the domain and range of a function?

A: To determine the domain and range of a function, you need to follow these steps:

1. Identify the type of function, such as linear, quadratic, or polynomial.
2. Determine the values of the independent variable that make the function undefined.
3. Determine the values of the dependent variable that the function can take.
4. Write the domain and range in interval notation.

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

A: A rational function is a function that can be written in the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, while an irrational function is a function that cannot be written in this form. For example, the function $f(x) = \frac{x^2 + 1}{x + 1}$ is a rational function, while the function $f(x) = \sqrt{x^2 + 1}$ is an irrational function.

Q: How do I simplify a rational function?

A: To simplify a rational function, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the expression by applying the rules of arithmetic operations.

Q: What is the difference between a polynomial function and a non-polynomial function?

A: A polynomial function is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$, while a non-polynomial function is a function that cannot be written in this form. For example, the function $f(x) = x^2 + 2x + 1$ is a polynomial function, while the function $f(x) = \sin x$ is a non-polynomial function.

Q: How do I determine the degree of a polynomial function?

A: To determine the degree of a polynomial function, you need to follow these steps:

1. Identify the highest power of the variable in the polynomial.
2. Write the degree as a positive integer.

Q: What is the difference between a monomial and a polynomial?

A: A monomial is a single term that consists of a coefficient and a variable, while a polynomial is a sum of monomials. For example, the expression $2x + 3$ is a polynomial, while the expression $2x$ is a monomial.

Q: How do I simplify a polynomial function?

A: To simplify a polynomial function, you need to follow these steps:

1. Combine like terms by adding or subtracting the coefficients of the same variables.
2. Simplify the expression by applying the rules of arithmetic operations.

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

A: A linear polynomial is a polynomial of degree 1, while a quadratic polynomial is a polynomial of degree 2. For example, the polynomial $2x + 3$ is a linear polynomial, while the polynomial $x^2 + 2x + 1$ is a quadratic polynomial.

Q: How do I determine the leading coefficient of a polynomial function?

A: To determine the leading coefficient of a polynomial function, you need to follow these steps:

1. Identify the highest power of the variable in the polynomial.
2. Write the leading coefficient as the coefficient of the highest power of the variable.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$, while a rational function is a function that can be written in the form $f(x) = \frac{p(x