Solve \( \frac{x}{3}-\frac{1}{4}(x+2)>3 x-2 \frac{1}{5} \), illustrate the solution on a number line.

To solve the inequality \( \frac{x}{3} - \frac{1}{4}(x+2) > 3x - 2 \frac{1}{5} \), let’s start by simplifying both sides. First, simplify the left side: \[ \frac{x}{3} - \frac{1}{4}(x + 2) = \frac{x}{3} - \frac{x}{4} - \frac{1}{2} \] To combine \( \frac{x}{3} \) and \( \frac{x}{4} \), we need a common denominator, which is 12: \[ \frac{x}{3} = \frac{4x}{12}, \quad \frac{x}{4} = \frac{3x}{12} \] So, \[ \frac{x}{3} - \frac{x}{4} = \frac{4x}{12} - \frac{3x}{12} = \frac{x}{12} \] Thus, we have: \[ \frac{x}{12} - \frac{1}{2} > 3x - 2 \frac{1}{5} \] Now, simplify the right side: \[ 2 \frac{1}{5} = \frac{11}{5}, \quad \text{so, } 3x - 2 \frac{1}{5} = 3x - \frac{11}{5} \] Now, we can rewrite our inequality as: \[ \frac{x}{12} - \frac{1}{2} > 3x - \frac{11}{5} \] Next, let’s eliminate the fractions by multiplying the entire inequality by 60 (the least common multiple of 12, 2, and 5): \[ 60 \left(\frac{x}{12}\right) - 60 \left(\frac{1}{2}\right) > 60 \left(3x - \frac{11}{5}\right) \] This simplifies to: \[ 5x - 30 > 180x - 132 \] Now, combine like terms: \[ 5x + 132 > 180x + 30 \] Rearranging gives: \[ 132 - 30 > 180x - 5x \] \[ 102 > 175x \] Dividing by 175: \[ \frac{102}{175} > x \] Or, equivalently: \[ x < \frac{102}{175} \] Now, to illustrate this on a number line, you would draw a line and put an open circle at \( \frac{102}{175} \) (approximately 0.583) to indicate that this value is not included in the solution. Now, on the number line, everything to the left of \( \frac{102}{175} \) will be shaded, representing all the valid \( x \) values for the inequality. In summary, the solution to the inequality \( x < \frac{102}{175} \) represents the set of numbers that satisfy the original inequality.