The given inequality is:

Question

ScanSolve AI Tutor · Accepted Answer

To solve the inequality, we will first find a common denominator for all terms and then clear the fractions.

The given inequality is:

-1 + \frac{2}{3p} \le \frac{1}{4} - \frac{5}{6p}

We are given that $p > 0$ .

Find the least common multiple (LCM) of the denominators $1, 3p, 4,$ and $6p$.
The LCM of $1, 3p, 4, 6p$ is $12p$.
Multiply every term in the inequality by the LCM, $12p$.
Since $p > 0$ , $12p$ is positive, so the inequality sign will not change direction.
$12p(-1) + 12p\left(\frac{2}{3p}\right) \le 12p\left(\frac{1}{4}\right) - 12p\left(\frac{5}{6p}\right)$
Simplify each term.
$-12p + \frac{24p}{3p} \le \frac{12p}{4} - \frac{60p}{6p}$ $-12p + 8 \le 3p - 10$
Gather terms with $p$ on one side and constant terms on the other side.
Add $12p$ to both sides:
$8 \le 3p + 12p - 10$ $8 \le 15p - 10$
Add 10 to both sides:
$8 + 10 \le 15p$ $18 \le 15p$
Solve for $p$ .
Divide both sides by 15:
$\frac{18}{15} \le p$
Simplify the fraction.
$\frac{6}{5} \le p$
This can also be written as $p \ge \frac{6}{5}$ .
Since $\frac{6}{5} = 1.2$ , and $1.2 > 0$ , the condition $p > 0$ is satisfied.

The solution for $p$ is:

Answer