https://r2.erweima.ai/mathbot/image/172675364540262paflxz.jpg

This image shows four equations where the variable \( x \) is present in the denominator, and the goal is to solve for \( x \). Let's work through each of them:

### 1. Equation 1 (Top-Right)
\[
4 + \frac{1}{x} = 8
\]
Subtract 4 from both sides:
\[
\frac{1}{x} = 8 - 4
\]
\[
\frac{1}{x} = 4
\]
Now, take the reciprocal of both sides:
\[
x = \frac{1}{4}
\]

### 2. Equation 2 (Bottom-Left)
\[
3 + \frac{1}{x} = 7
\]
Subtract 3 from both sides:
\[
\frac{1}{x} = 7 - 3
\]
\[
\frac{1}{x} = 4
\]
Again, take the reciprocal:
\[
x = \frac{1}{4}
\]

### 3. Equation 3 (Bottom-Right)
\[
2 + \frac{1}{x} = 5
\]
Subtract 2 from both sides:
\[
\frac{1}{x} = 5 - 2
\]
\[
\frac{1}{x} = 3
\]
Take the reciprocal:
\[
x = \frac{1}{3}
\]

### 4. Equation 4 (Top-Left)
\[
X = ?
\]

Now, based on these results, we can conclude that \( x = \frac{1}{4} \) satisfies the first two equations, while \( x = \frac{1}{3} \) satisfies the third equation.

Would you like further details or clarifications on these steps?

Here are 5 questions to explore further:
1. What happens if you have a similar equation but with a negative constant?
2. How would you handle equations like this if they involve fractions?
3. Can these equations be solved graphically? What would the graph look like?
4. What is the general approach for solving equations with variables in the denominator?
5. How would these equations change if we introduce a coefficient for \( x \)?

**Tip**: Always check for restrictions on \( x \) in denominators to avoid division by zero.

Solve for X in the following set of equations: 4 + 1/x = 8, 3 + 1/x = 7, 2 + 1/x = 5.

Learn how to solve algebraic equations where the variable X is in the denominator. The step-by-step solution includes solving 4 + 1/x = 8, 3 + 1/x = 7, and 2 + 1/x = 5, perfect for students in Grades 6-8.

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