Answer
The solution is $\Big(-\infty,\dfrac{26}{9}\Big)\cup\Big(\dfrac{34}{9},\infty\Big)$
Work Step by Step
$\Big|\dfrac{5}{3}-\dfrac{1}{2}x\Big|\gt\dfrac{2}{9}$
Solving this absolute value inequality is equivalent to solving two separate inequalities, which are:
$\dfrac{5}{3}-\dfrac{1}{2}x\gt\dfrac{2}{9}$ and $\dfrac{5}{3}-\dfrac{1}{2}x\lt-\dfrac{2}{9}$
$\textbf{Solve the first inequality:}$
$\dfrac{5}{3}-\dfrac{1}{2}x\gt\dfrac{2}{9}$
Multiply the whole inequality by $18$:
$18\Big(\dfrac{5}{3}-\dfrac{1}{2}x\gt\dfrac{2}{9}\Big)$
$30-9x\gt4$
Take $30$ to the right side:
$-9x\gt4-30$
$-9x\gt-26$
Take $-9$ to divide the right side and reverse the direction of the inequality sign:
$x\lt\dfrac{-26}{-9}$
$x\lt\dfrac{26}{9}$
$\textbf{Solve the second inequality:}$
$\dfrac{5}{3}-\dfrac{1}{2}x\lt-\dfrac{2}{9}$
Multiply the whole inequality by $18$:
$18\Big(\dfrac{5}{3}-\dfrac{1}{2}x\lt-\dfrac{2}{9}\Big)$
$30-9x\lt-4$
Take $30$ to the right side:
$-9x\lt-4-30$
$-9x\lt-34$
Take $-9$ to divide the right side and reverse the direction of the inequality sign:
$x\gt\dfrac{-34}{-9}$
$x\gt\dfrac{34}{9}$
Expressing the solution in interval notation:
$\Big(-\infty,\dfrac{26}{9}\Big)\cup\Big(\dfrac{34}{9},\infty\Big)$
