Linear Inequalities

Concept:Solve a linear inequality by simplifying, combining like terms, and isolating the variable.Explanation:First, rewrite the mixed fraction: $1\frac{1}{3} = \frac{4}{3}$.So the inequality is $\frac{x}{3} - \frac{1}{4}(x+2) > 3x - \frac{4}{3}$.Combine the left side over a common denominator 12:$\frac{x}{3} - \frac{x+2}{4} = \frac{4x - 3(x+2)}{12} = \frac{4x - 3x - 6}{12} = \frac{x-6}{12}$.Now we have $\frac{x-6}{12} > 3x - \frac{4}{3}$.Multiply both sides by 12 (positive, so inequality direction stays the same):$x - 6 > 36x - 16$.Bring $x$ terms to one side and constants to the other:$x - 36x > -16 + 6$ ⟹ $-35x > -10$.Multiply both sides by $-1$ (remember to reverse the inequality sign):$35x < 10$ ⟹ $x < \frac{10}{35} = \frac{2}{7}$.Answer:$x < \frac{2}{7}$