What values of the variable cannot possibly be solutions for the given​ equation, without actually solving the​ equation?
StartFraction 8 Over 2 x plus 5 EndFraction minus StartFraction 1 Over x minus 6 EndFraction equals 0
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.


To find the values of the variable that **cannot possibly be solutions** for the given equation, we need to focus on the denominators of the rational expressions. Any value of \( x \) that makes a denominator zero is not a valid solution, because division by zero is undefined.

Given the equation:

\[
\frac{8}{2x + 5} - \frac{1}{x - 6} = 0
\]

### Step 1: Identify the denominators
The denominators are \( 2x + 5 \) and \( x - 6 \).

### Step 2: Set each denominator equal to zero and solve for \( x \)

1. \( 2x + 5 = 0 \)
   \[
   2x = -5
   \]
   \[
   x = -\frac{5}{2}
   \]
   
2. \( x - 6 = 0 \)
   \[
   x = 6
   \]

### Step 3: Conclusion
The values \( x = -\frac{5}{2} \) and \( x = 6 \) make the denominators zero, and therefore, these values cannot possibly be solutions to the equation.

### Final Answer:
The values of \( x \) that cannot be solutions are:
\[
x = -\frac{5}{2}, \, x = 6
\]

Would you like further details or have any questions?

### Relative questions:
1. How can we verify whether a value makes a denominator zero?
2. Why is division by zero undefined in mathematics?
3. Can this method be applied to all rational equations?
4. How would the equation change if there were no restrictions on the values of \( x \)?
5. What are the next steps to actually solve the equation?

### Tip:
Always check the denominators first in any rational equation to avoid undefined expressions.

What values of the variable cannot possibly be solutions for the given equation, without actually solving the equation? (8 / (2x + 5)) - (1 / (x - 6)) = 0

Learn how to identify values of x that cannot be solutions for the rational equation (8 / (2x + 5)) - (1 / (x - 6)) = 0. This problem demonstrates how to set denominators equal to zero to avoid undefined values. Suitable for students in Grades 9-11.

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