If `x/4 lt (5x-2)/3 -(7x-3)/5, x in R` then

Options

A. `x in (- oo,4]`
B. `x in (- oo,4)`
C. `x in (4,oo)`
D. `x in [4,oo)`

Answer

To solve the inequality \( \frac{x}{4} < \frac{5x - 2}{3} - \frac{7x - 3}{5} \), we will follow these steps:

### Step 1: Rewrite the inequality
Start with the given inequality:
\[ \frac{x}{4} < \frac{5x - 2}{3} - \frac{7x - 3}{5} \]

### Step 2: Find a common denominator
The denominators are 4, 3, and 5. The least common multiple (LCM) of these numbers is 60. We will rewrite each term with a denominator of 60.

### Step 3: Rewrite each fraction
- For \( \frac{x}{4} \):
\[ \frac{x}{4} = \frac{15x}{60} \]
- For \( \frac{5x - 2}{3} \):
\[ \frac{5x - 2}{3} = \frac{20(5x - 2)}{60} = \frac{100x - 40}{60} \]
- For \( \frac{7x - 3}{5} \):
\[ \frac{7x - 3}{5} = \frac{12(7x - 3)}{60} = \frac{84x - 36}{60} \]

### Step 4: Substitute back into the inequality
Now substitute these back into the inequality:
\[ \frac{15x}{60} < \frac{100x - 40}{60} - \frac{84x - 36}{60} \]

### Step 5: Combine the fractions on the right
Combine the right-hand side:
\[ \frac{100x - 40 - 84x + 36}{60} = \frac{(100x - 84x) + (-40 + 36)}{60} = \frac{16x - 4}{60} \]

### Step 6: Set up the new inequality
Now we have:
\[ \frac{15x}{60} < \frac{16x - 4}{60} \]

### Step 7: Eliminate the denominator
Since the denominators are the same, we can multiply through by 60 (which is positive, so the inequality direction remains the same):
\[ 15x < 16x - 4 \]

### Step 8: Rearrange the inequality
Now, rearranging gives:
\[ 15x - 16x < -4 \implies -x < -4 \]

### Step 9: Multiply by -1
Multiplying both sides by -1 (remember to flip the inequality):
\[ x > 4 \]

### Step 10: Write the solution in interval notation
The solution in interval notation is:
\[ (4, \infty) \]

### Conclusion
Thus, the solution to the inequality is:
\[ x \in (4, \infty) \]