Question

54) |1-2/3x|<1

Answer 1

$|a| < b\iff a < b\ \wedge\ a > -b\\\\|a| > b\iff a > b\ \vee\ a < -b$

$\left|1-\dfrac{2}{3}x\right| < 1\iff1-\dfrac{2}{3}x < 1\ \wedge\ 1-\dfrac{2}{3}x > -1\ \ \ |\text{subtract 1 from both sides}\\\\-\dfrac{2}{3}x < 0\ \vedge\ -\dfrac{2}{3}x > -2\ \ \ \ |\text{change the signs}\\\\\dfrac{2}{3}x > 0\ \wedge\ \dfrac{2}{3}x < 2\ \ \ \ |\text{multiply both sides by 3}\\\\2x > 0\ \wedge\ 2x < 6\ \ \ \ |\text{divide both sides by 2}\\\\x > 0\ \wedge\ x < 3\to0 < x < 3\to x\in(0,\ 3)$