Solve the inequalities for real x and represent solution on number line
`(x-3)/(x-5)>2`

Answer

To solve the inequality \(\frac{x-3}{x-5} > 2\), we will follow these steps:

### Step 1: Rewrite the Inequality
Start by rewriting the inequality:
\[ \frac{x-3}{x-5} - 2 > 0 \]

### Step 2: Combine the Terms
To combine the terms on the left side, we need a common denominator:
\[ \frac{x-3 - 2(x-5)}{x-5} > 0 \]
This simplifies to:
\[ \frac{x-3 - 2x + 10}{x-5} > 0 \]

### Step 3: Simplify the Numerator
Now, simplify the numerator:
\[ \frac{-x + 7}{x-5} > 0 \]
or equivalently,
\[ \frac{7 - x}{x-5} > 0 \]

### Step 4: Identify Critical Points
Next, we need to find the critical points where the expression is equal to zero or undefined. Set the numerator and denominator to zero:
1. \(7 - x = 0 \Rightarrow x = 7\)
2. \(x - 5 = 0 \Rightarrow x = 5\)

### Step 5: Test Intervals
Now, we will test the intervals determined by the critical points \(x = 5\) and \(x = 7\):
- Interval 1: \( (-\infty, 5) \)
- Interval 2: \( (5, 7) \)
- Interval 3: \( (7, \infty) \)

**Test Interval 1: \(x = 0\)**
\[ \frac{7 - 0}{0 - 5} = \frac{7}{-5} < 0 \quad \text{(not a solution)} \]

**Test Interval 2: \(x = 6\)**
\[ \frac{7 - 6}{6 - 5} = \frac{1}{1} > 0 \quad \text{(solution)} \]

**Test Interval 3: \(x = 8\)**
\[ \frac{7 - 8}{8 - 5} = \frac{-1}{3} < 0 \quad \text{(not a solution)} \]

### Step 6: Write the Solution
From our testing, the solution to the inequality is:
\[ x \in (5, 7) \]

### Step 7: Represent on Number Line
On the number line, we will represent the solution as follows:
- Draw a number line.
- Mark points 5 and 7.
- Use open circles at 5 and 7 to indicate that these points are not included in the solution.
- Shade the region between 5 and 7.

### Final Solution
Thus, the solution to the inequality \(\frac{x-3}{x-5} > 2\) is:
\[ x \in (5, 7) \]

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