Solve the given inequality for real \(x:\frac{ (2x -1)}{3} ≥ \frac{(3x -2)}{4} - \frac{(2-x)}{5}\).
Given Inequality:
\( \frac{2x - 1}{3} \ge \frac{3x - 2}{4} - \frac{2 - x}{5} \)
Step 1: Take LCM of denominators (3, 4, 5)
LCM = 60
Multiply both sides by 60:
\( 20(2x - 1) \ge 15(3x - 2) - 12(2 - x) \)
Step 2: Simplify both sides
Left-hand side:
20(2x − 1) = 40x − 20
Right-hand side:
15(3x − 2) = 45x − 30
12(2 − x) = 24 − 12x
So,
Right-hand side = 45x − 30 − 24 + 12x = 57x − 54
Step 3: Form the inequality
40x − 20 ≥ 57x − 54
Step 4: Solve the inequality
40x − 57x ≥ −54 + 20
−17x ≥ −34
Dividing by −17 (inequality sign reverses):
\( x \le 2 \)
Graphical Representation on Number Line:
A closed circle at 2 and the region to the left of 2 is shaded.
Final Answer:
The solution set is
{ x ∣ x ≤ 2 }