Solve the given inequality for real\(x:\frac{ x}{3} > \frac{x}{2} +1\).
Given Inequality:
\(\frac{x}{3} > \frac{x}{2} + 1\)
Step 1: Eliminate fractions
Multiply both sides by LCM of 3 and 2, i.e., 6:
\(2x > 3x + 6\)
Step 2: Solve the inequality
2x − 3x > 6
−x > 6
Dividing by −1 (inequality sign reverses):
\(x < -6\)
Graphical Representation on Number Line:
An open circle at −6 and the region to the left of −6 is shaded.
Final Answer:
The solution set is
{ x ∣ x < −6 }