Solve the given inequality for real\(x: \frac{x}{4} < \frac{(5x-2)}{3} -\frac{ (7x-3)}{5}\).
Given Inequality:
\(\frac{x}{4} < \frac{5x-2}{3} - \frac{7x-3}{5}\)
Step 1: Take LCM of denominators (4, 3, 5)
LCM = 60
Multiply both sides by 60:
\(15x < 20(5x-2) - 12(7x-3)\)
Step 2: Simplify the right-hand side
20(5x − 2) = 100x − 40
12(7x − 3) = 84x − 36
So,
Right-hand side = 100x − 40 − 84x + 36 = 16x − 4
Step 3: Form the inequality
15x < 16x − 4
Step 4: Solve the inequality
15x − 16x < −4
−x < −4
Dividing by −1 (inequality sign reverses):
\(x > 4\)
Graphical Representation on Number Line:
An open circle at 4 and the region to the right of 4 is shaded.
Final Answer:
The solution set is
{ x ∣ x > 4 }