College Algebra, Chapter 1, 1.6, Section 1.6, Problem 24

Solve the linear inequality $\displaystyle \frac{2}{3} - \frac{1}{2} x \geq \frac{1}{6} + x$. Express the solution using interval notation and graph the solution set.


$
\begin{equation}
\begin{aligned}

& \frac{2}{3} - \frac{1}{2} x \geq \frac{1}{6} + x
&& \text{Given}
\\
\\
& \frac{2}{3} - \frac{1}{6} \geq x + \frac{1}{2} x
&& \text{Add } \frac{1}{2} x \text{ and subtract } \frac{1}{6}
\\
\\
& \frac{4 - 1}{6} \geq \frac{3}{2} x
&& \text{Common denominator}
\\
\\
& \frac{3}{6} \geq \frac{3}{2} x
&& \text{Simplify}
\\
\\
& \frac{3}{6} \left( \frac{2}{3} \right) \geq x
&& \text{Multiply both sides by } \frac{2}{3}
\\
\\
& \frac{1}{3} \geq x
&& \text{Simplify}

\end{aligned}
\end{equation}
$