Solve the inequalities for real x : `x/4<((5x-2))/3-((7x-3))/5`

Explanation

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 original inequality:
\[ \frac{x}{4} < \frac{5x - 2}{3} - \frac{7x - 3}{5} \]

### Step 2: Find a common denominator
To combine the fractions on the right side, we need to find a common denominator. The least common multiple of 3 and 5 is 15. Thus, we rewrite the fractions:
\[ \frac{5x - 2}{3} = \frac{5(5x - 2)}{15} = \frac{25x - 10}{15} \]
\[ \frac{7x - 3}{5} = \frac{3(7x - 3)}{15} = \frac{21x - 9}{15} \]

### Step 3: Combine the fractions
Now, substitute back into the inequality:
\[ \frac{x}{4} < \frac{25x - 10 - (21x - 9)}{15} \]
Simplifying the right side:
\[ 25x - 10 - 21x + 9 = 4x - 1 \]
So the inequality becomes:
\[ \frac{x}{4} < \frac{4x - 1}{15} \]

### Step 4: Cross multiply
To eliminate the fractions, we cross multiply:
\[ 15x < 4(4x - 1) \]
Expanding the right side:
\[ 15x < 16x - 4 \]

### Step 5: Rearrange the inequality
Now, we rearrange the inequality:
\[ 15x - 16x < -4 \]
This simplifies to:
\[ -x < -4 \]

### Step 6: Solve for x
Multiplying both sides by -1 (remember to flip the inequality sign):
\[ x > 4 \]

### Step 7: Write the solution
The solution to the inequality is:
\[ x \in (4, \infty) \]