Answer to Question #348807 in Real Analysis for Dhruv bartwal

Question

Show that if 5/3< 2x< 11/3, then x∈{y∈R such that |y- 4/3| < 1/2}.

Answer 1

$$\dfrac{5}{3}<2x<\dfrac{11}{3}$$

$$\dfrac{5}{6}<x<\dfrac{11}{6}$$

$$\dfrac{5}{6}-\dfrac{4}{3}<x-\dfrac{4}{3}<\dfrac{11}{6}-\dfrac{4}{3}$$

$$-\dfrac{1}{2}<x-\dfrac{4}{3}<\dfrac{1}{2}$$

Then

$$|x-\dfrac{4}{3}|<\dfrac{1}{2}, if\ \dfrac{5}{3}<2x<\dfrac{11}{3}$$