Chapter 8.8, Problem 53IP

To determine

Solve the inequality $p>\frac{2}{3}\left(p-\frac{1}{2}\right)$

Answer to Problem 53IP

The solution of the inequality is $p>1$

Explanation of Solution

Given:

The inequality, $p>\frac{2}{3}\left(p-\frac{1}{2}\right)$

Concept used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• Inequality sign get reverse on multiplying or dividing by a negative number

Calculation:

• In order to solve the mixed inequality $p>\frac{2}{3}\left(p-\frac{1}{2}\right)$ first isolate the variable term p by performing some basic operations. Here to isolate p terms on the left side, first multiply both sides by $3$ and then divide both sides by $2$ and then add both sides by $p$ and then simplify further as shown below:

  $\begin{array}{c}p>\frac{2}{3}\left(p-\frac{1}{2}\right)\\ p\times 3>\frac{2}{3}\left(p-\frac{1}{2}\right)\times 3\\ 3p>2\left(p-\frac{1}{2}\right)\\ \frac{3p}{2}>\frac{2}{2}\left(p-\frac{1}{2}\right)\\ \frac{3p}{2}>p-\frac{1}{2}\\ \frac{3p}{2}-p\times \frac{2}{2}>\frac{1}{2}\\ \frac{3p}{2}-\frac{2p}{2}>\frac{1}{2}\\ \frac{p}{2}>\frac{1}{2}\\ \frac{p}{2}\times 2>\frac{1}{2}\times 2\\ p>1\end{array}$

Thus, the solution of the given inequality is $p>1$ .