4x-x/2<0

To solve the inequality \( 4x - \frac{x}{2} < 0 \), we first simplify the left side. 1. Rewrite \(\frac{x}{2}\) so that it has a common denominator with \(4x\): \[ 4x = \frac{8x}{2} \] 2. Now substitute this back into the inequality: \[ \frac{8x}{2} - \frac{x}{2} < 0 \] 3. Combine the fractions: \[ \frac{8x - x}{2} < 0 \] \[ \frac{7x}{2} < 0 \] 4. Now, to eliminate the fraction, multiply both sides of the inequality by 2 (note that since 2 is positive, the direction of the inequality does not change): \[ 7x < 0 \] 5. Finally, divide both sides by 7: \[ x < 0 \] Thus, the solution to the inequality \( 4x - \frac{x}{2} < 0 \) is: \[ \boxed{x < 0} \]