The complete set of values of 'x' which satisfy the inequations: `5x+2<3x+8` and `(x+2)/(x-1)<4` is :

Options

A. `(-oo , 1) cup ( 2,3)`
B. `(0,oo)`
C. `(-oo,2)`
D. `(1,3)`

Answer

To solve the given inequalities \(5x + 2 < 3x + 8\) and \(\frac{x + 2}{x - 1} < 4\), we will solve each inequality step by step and then find the intersection of the solutions.

### Step 1: Solve the first inequality \(5x + 2 < 3x + 8\)

1. Start by rearranging the inequality:
\[ 5x + 2 - 3x < 8 \]
This simplifies to:
\[ 2x + 2 < 8 \]

2. Next, subtract 2 from both sides:
\[ 2x < 6 \]

3. Finally, divide both sides by 2:
\[ x < 3 \]

### Step 2: Solve the second inequality \(\frac{x + 2}{x - 1} < 4\)

1. Start by rearranging the inequality:
\[ \frac{x + 2}{x - 1} - 4 < 0 \]

2. Rewrite \(4\) as \(\frac{4(x - 1)}{x - 1}\):
\[ \frac{x + 2 - 4(x - 1)}{x - 1} < 0 \]
This simplifies to:
\[ \frac{x + 2 - 4x + 4}{x - 1} < 0 \]
Which further simplifies to:
\[ \frac{-3x + 6}{x - 1} < 0 \]
Or:
\[ \frac{3(2 - x)}{x - 1} < 0 \]

3. Now, we need to find the critical points by setting the numerator and denominator to zero:
- The numerator \(3(2 - x) = 0\) gives \(x = 2\).
- The denominator \(x - 1 = 0\) gives \(x = 1\).

4. We will test intervals around the critical points \(x = 1\) and \(x = 2\):
- For \(x < 1\) (e.g., \(x = 0\)): \(\frac{3(2 - 0)}{0 - 1} > 0\) (positive)
- For \(1 < x < 2\) (e.g., \(x = 1.5\)): \(\frac{3(2 - 1.5)}{1.5 - 1} < 0\) (negative)
- For \(x > 2\) (e.g., \(x = 3\)): \(\frac{3(2 - 3)}{3 - 1} < 0\) (negative)

5. Therefore, the solution for the second inequality is:
\[ 1 < x < 2 \quad \text{or} \quad x > 2 \]

### Step 3: Combine the solutions

From the first inequality, we have:
\[ x < 3 \]

From the second inequality, we have:
\[ 1 < x < 2 \quad \text{or} \quad x > 2 \]

Now, we need to find the intersection of these two sets:
1. For \(1 < x < 2\), this is entirely within \(x < 3\).
2. For \(x > 2\), we have \(2 < x < 3\).

Thus, the complete set of values of \(x\) that satisfy both inequalities is:
\[ x \in (1, 2) \cup (2, 3) \]

### Final Answer:
The complete set of values of \(x\) which satisfy the inequalities is:
\[ (1, 3) \]