(b) Solve the inequality \( \left|3-\frac{1}{x}\right|<\frac{1}{2} \) and give your answer in interval notation.
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Mind Expander
To solve the inequality \( \left|3-\frac{1}{x}\right|<\frac{1}{2} \), we split it into two cases based on the definition of absolute value: 1. \( 3 - \frac{1}{x} < \frac{1}{2} \) 2. \( 3 - \frac{1}{x} > -\frac{1}{2} \) ### Case 1: \( 3 - \frac{1}{x} < \frac{1}{2} \) Rearranging it, we have: \[ -\frac{1}{x} < \frac{1}{2} - 3 \] \[ -\frac{1}{x} < -\frac{5}{2} \] Multiplying both sides by -1 (which flips the inequality): \[ \frac{1}{x} > \frac{5}{2} \] This gives: \[ x < \frac{2}{5} \] ### Case 2: \( 3 - \frac{1}{x} > -\frac{1}{2} \) Rearranging this inequality: \[ -\frac{1}{x} > -\frac{1}{2} - 3 \] \[ -\frac{1}{x} > -\frac{7}{2} \] Again, multiplying by -1 (flipping the inequality): \[ \frac{1}{x} < \frac{7}{2} \] This leads to: \[ x > \frac{2}{7} \] ### Combine the results Thus, we have two conditions: 1. \( x < \frac{2}{5} \) 2. \( x > \frac{2}{7} \) Since we are looking for solutions where both conditions hold, the solution in interval notation becomes: \[ \left(\frac{2}{7}, \frac{2}{5}\right) \] This is your final answer!
