For Which Rational Expression Is − 5 -5 − 5 An Excluded Value Of X X X ?A. X − 5 6 \frac{x-5}{6} 6 X − 5 ​ B. X + 5 6 \frac{x+5}{6} 6 X + 5 ​ C. 6 X − 5 \frac{6}{x-5} X − 5 6 ​ D. 6 X + 5 \frac{6}{x+5} X + 5 6 ​

Understanding Excluded Values

In mathematics, a rational expression is a fraction that contains variables in the numerator or denominator. When working with rational expressions, it's essential to identify the values of the variable that would make the expression undefined. These values are known as excluded values.

What are Excluded Values?

Excluded values occur when the denominator of a rational expression is equal to zero. This is because division by zero is undefined in mathematics. Therefore, any value of the variable that makes the denominator zero is an excluded value.

Finding Excluded Values

To find the excluded values of a rational expression, we need to set the denominator equal to zero and solve for the variable. This will give us the values that make the expression undefined.

Applying to the Given Options

Let's apply this concept to the given options and find the excluded values for each expression.

Option A: $\frac{x-5}{6}$

To find the excluded value for this expression, we need to set the denominator equal to zero and solve for $x$.

$$\frac{x-5}{6} = 0$$

Multiplying both sides by 6, we get:

$$x - 5 = 0$$

Adding 5 to both sides, we get:

$$x = 5$$

Therefore, the excluded value for option A is $x = 5$.

Option B: $\frac{x+5}{6}$

To find the excluded value for this expression, we need to set the denominator equal to zero and solve for $x$.

$$\frac{x+5}{6} = 0$$

Multiplying both sides by 6, we get:

$$x + 5 = 0$$

Subtracting 5 from both sides, we get:

$$x = -5$$

Therefore, the excluded value for option B is $x = -5$.

Option C: $\frac{6}{x-5}$

To find the excluded value for this expression, we need to set the denominator equal to zero and solve for $x$.

$$\frac{6}{x-5} = 0$$

Since the numerator is a constant, the only way for the expression to be equal to zero is if the denominator is equal to zero.

$$x - 5 = 0$$

Adding 5 to both sides, we get:

$$x = 5$$

Therefore, the excluded value for option C is $x = 5$.

Option D: $\frac{6}{x+5}$

To find the excluded value for this expression, we need to set the denominator equal to zero and solve for $x$.

$$\frac{6}{x+5} = 0$$

Since the numerator is a constant, the only way for the expression to be equal to zero is if the denominator is equal to zero.

$$x + 5 = 0$$

Subtracting 5 from both sides, we get:

$$x = -5$$

Therefore, the excluded value for option D is $x = -5$.

Conclusion

In conclusion, the excluded value of $x$ for the given rational expressions is:

• Option A: $x = 5$
• Option B: $x = -5$
• Option C: $x = 5$
• Option D: $x = -5$

Therefore, the correct answer is option B, $\frac{x+5}{6}$.

Final Answer

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Q: What is a rational expression?

A: A rational expression is a fraction that contains variables in the numerator or denominator.

Q: What is an excluded value?

A: An excluded value is a value of the variable that makes the denominator of a rational expression equal to zero, making the expression undefined.

Q: How do I find the excluded values of a rational expression?

A: To find the excluded values of a rational expression, set the denominator equal to zero and solve for the variable.

Q: What happens when the denominator of a rational expression is equal to zero?

A: When the denominator of a rational expression is equal to zero, the expression is undefined, and the value of the variable that makes the denominator zero is an excluded value.

Q: Can a rational expression have more than one excluded value?

A: Yes, a rational expression can have more than one excluded value. This occurs when the denominator of the expression is a product of two or more factors, and each factor can be equal to zero.

Q: How do I simplify a rational expression with excluded values?

A: To simplify a rational expression with excluded values, factor the numerator and denominator, and cancel out any common factors. However, be careful not to cancel out any factors that would make the denominator equal to zero.

Q: Can I add or subtract rational expressions with excluded values?

A: Yes, you can add or subtract rational expressions with excluded values, but you must first find a common denominator and then add or subtract the numerators.

Q: Can I multiply or divide rational expressions with excluded values?

A: Yes, you can multiply or divide rational expressions with excluded values, but you must first find a common denominator and then multiply or divide the numerators.

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

A: A rational number is a number that can be expressed as the ratio of two integers, while a rational expression is a fraction that contains variables in the numerator or denominator.

Q: Can a rational expression be a rational number?

A: Yes, a rational expression can be a rational number if the numerator and denominator are integers and the denominator is not equal to zero.

Q: Can a rational number be a rational expression?

A: Yes, a rational number can be a rational expression if the numerator and denominator are variables and the denominator is not equal to zero.

Q: What is the importance of understanding excluded values in rational expressions?

A: Understanding excluded values in rational expressions is crucial because it helps you identify the values of the variable that make the expression undefined, which is essential in solving equations and inequalities involving rational expressions.

Q: Can you provide examples of rational expressions with excluded values?

A: Yes, here are some examples of rational expressions with excluded values:

• $\frac{x-5}{6}$
• $\frac{x+5}{6}$
• $\frac{6}{x-5}$
• $\frac{6}{x+5}$

Q: Can you provide examples of rational expressions without excluded values?

A: Yes, here are some examples of rational expressions without excluded values:

• $\frac{x}{6}$
• $\frac{6}{x}$
• $\frac{x+5}{x-5}$
• $\frac{x-5}{x+5}$

Q: Can you provide examples of rational numbers that are also rational expressions?

A: Yes, here are some examples of rational numbers that are also rational expressions:

• $\frac{1}{2}$
• $\frac{3}{4}$
• $\frac{2}{3}$
• $\frac{5}{6}$

Q: Can you provide examples of rational expressions that are not rational numbers?

A: Yes, here are some examples of rational expressions that are not rational numbers:

• $\frac{x}{6}$
• $\frac{6}{x}$
• $\frac{x+5}{x-5}$
• $\frac{x-5}{x+5}$