2. $\frac{-9}{4}(x - 6) < \frac{1}{9}(6 - x)$

Explanation:

Step1: Expand both sides

$-\frac{9}{4}x+\frac{9}{4}\times6<\frac{1}{9}\times6-\frac{1}{9}x$ $-\frac{9}{4}x+\frac{27}{2}<\frac{2}{3}-\frac{1}{9}x$

Step2: Move terms with $x$ to one - side

$-\frac{9}{4}x+\frac{1}{9}x<\frac{2}{3}-\frac{27}{2}$

Step3: Find a common denominator for $x$ terms and constant terms

For $x$ terms: The common denominator of 4 and 9 is 36. $-\frac{9}{4}x+\frac{1}{9}x=-\frac{81}{36}x+\frac{4}{36}x=-\frac{77}{36}x$. For constant terms: The common denominator of 3 and 2 is 6. $\frac{2}{3}-\frac{27}{2}=\frac{4 - 81}{6}=-\frac{77}{6}$. So, $-\frac{77}{36}x<-\frac{77}{6}$.

Step4: Solve for $x$

Multiply both sides by $-\frac{36}{77}$. When multiplying an inequality by a negative number, the inequality sign flips. $x > -\frac{77}{6}\times(-\frac{36}{77})$ $x>6$

Answer:

$x > 6$