Number of integral value(s) of `m` for which the equation `sinx-sqrt(3)cosx=(4m-6)/(4-m)` has solutions, `x in [0,2pi]` , is ________

Explanation

To solve the problem, we need to find the number of integral values of \( m \) for which the equation

\[ \sin x - \sqrt{3} \cos x = \frac{4m - 6}{4 - m} \]

has solutions for \( x \) in the interval \( [0, 2\pi] \).

### Step 1: Determine the range of the left-hand side
The expression \( \sin x - \sqrt{3} \cos x \) can be rewritten in the form \( R \sin(x + \phi) \), where \( R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2 \). The maximum value of \( \sin x - \sqrt{3} \cos x \) is \( 2 \) and the minimum value is \( -2 \).

Thus, we have:

\[ -2 \leq \sin x - \sqrt{3} \cos x \leq 2 \]

### Step 2: Set up inequalities
We need to find when the right-hand side \( \frac{4m - 6}{4 - m} \) falls within this range. Therefore, we set up the inequalities:

\[ -2 \leq \frac{4m - 6}{4 - m} \leq 2 \]

### Step 3: Solve the left inequality
First, we solve the left inequality:

\[ -2 \leq \frac{4m - 6}{4 - m} \]

Multiplying both sides by \( 4 - m \) (noting that we need to consider the sign of \( 4 - m \)):

1. If \( 4 - m > 0 \) (i.e., \( m < 4 \)):
\[ -2(4 - m) \leq 4m - 6 \]
\[ -8 + 2m \leq 4m - 6 \]
\[ -8 + 6 \leq 4m - 2m \]
\[ -2 \leq 2m \implies m \geq -1 \]

2. If \( 4 - m < 0 \) (i.e., \( m > 4 \)):
\[ -2(4 - m) \geq 4m - 6 \]
\[ -8 + 2m \geq 4m - 6 \]
\[ -8 + 6 \geq 4m - 2m \]
\[ -2 \geq 2m \implies m \leq -1 \]
This case does not yield valid solutions since \( m \) cannot be both greater than 4 and less than or equal to -1.

Thus, from the left inequality, we have:

\[ m \geq -1 \quad \text{and} \quad m < 4 \]

### Step 4: Solve the right inequality
Now, we solve the right inequality:

\[ \frac{4m - 6}{4 - m} \leq 2 \]

Multiplying both sides by \( 4 - m \) (again considering the sign):

1. If \( 4 - m > 0 \) (i.e., \( m < 4 \)):
\[ 4m - 6 \leq 2(4 - m) \]
\[ 4m - 6 \leq 8 - 2m \]
\[ 4m + 2m \leq 8 + 6 \]
\[ 6m \leq 14 \implies m \leq \frac{14}{6} = \frac{7}{3} \]

2. If \( 4 - m < 0 \) (i.e., \( m > 4 \)):
\[ 4m - 6 \geq 2(4 - m) \]
This case leads to contradictions since it would imply \( m \) must be less than or equal to \( 4 \).

Thus, from the right inequality, we have:

\[ m \leq \frac{7}{3} \]

### Step 5: Combine the results
Combining the results from both inequalities, we have:

\[ -1 \leq m \leq \frac{7}{3} \]

### Step 6: Identify integral values
The integral values of \( m \) in the interval \( [-1, \frac{7}{3}] \) are:

- \( -1 \)
- \( 0 \)
- \( 1 \)
- \( 2 \)

Thus, the number of integral values of \( m \) is \( 4 \).

### Final Answer
The number of integral value(s) of \( m \) for which the equation has solutions in the interval \( [0, 2\pi] \) is **4**.