Math Snap

STEP 1

Assumptions1. The inequality is given by $\frac{5}{ p}+\frac{15}{6 p}-\frac{3+ p}{p} \geq3$
. $p>0$

STEP 2

First, we simplify the left side of the inequality. We can do this by finding a common denominator for the fractions.
$$\frac{5}{2 p}+\frac{15}{6 p}-\frac{+2 p}{p} = \frac{15}{6 p} + \frac{15}{6 p} - \frac{6 p +2 p^2}{2 p}$$

STEP 3

implify the fractions by combining the terms.
$$\frac{15}{6 p} + \frac{15}{6 p} - \frac{6 p +2 p^2}{2 p} = \frac{30}{6 p} - \frac{6 p +2 p^2}{2 p}$$

STEP 4

implify the fractions further.
$$\frac{30}{6 p} - \frac{6 p +2 p^2}{2 p} = \frac{}{p} - \frac{6 p +2 p^2}{2 p}$$

STEP 5

implify the fractions further.
$$\frac{5}{p} - \frac{ p +2 p^2}{2 p} = \frac{5}{p} -3 - p$$

STEP 6

Now we have the simplified inequality.
$$\frac{5}{p} -3 - p \geq3$$

STEP 7

Rearrange the inequality to isolate $p$.
$$\frac{5}{p} - p \geq6$$

STEP 8

Multiply through by $p$ to remove the fraction.
$$5 - p^2 \geq6p$$

STEP 9

Rearrange the inequality to form a quadratic inequality.
$$p^2 +6p -5 \leq$$

STEP 10

Factor the quadratic inequality.
$$(p -)(p +5) \leq0$$

STEP 11

olve the inequality. The solutions to the inequality are the values of $p$ that make the expression zero, and the sign of the inequality changes at these values.
So, $p - =0$ gives $p =$ and $p +5 =0$ gives $p = -5$.
However, we have the restriction that $p >0$, so we discard $p = -5$.

Test the intervals determined by the solutions to the inequality.For $p <$, $(p -)(p +5) >0$.
For $p >$, $(p -)(p +5) >0$.
So, the solution to the inequality is $p >$.
Therefore, the correct answer is $p \geq2$.