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QuestionSolve the inequality 52p+156p−3+2pp≥3\frac{5}{2 p}+\frac{15}{6 p}-\frac{3+2 p}{p} \geq 3 for p>0p > 0. What is pp?

Studdy Solution

STEP 1

Assumptions1. The inequality is given by 5p+156p−3+pp≥3\frac{5}{ p}+\frac{15}{6 p}-\frac{3+ p}{p} \geq3 . p>0p>0

STEP 2

First, we simplify the left side of the inequality. We can do this by finding a common denominator for the fractions.
52p+156p−+2pp=156p+156p−6p+2p22p\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.
156p+156p−6p+2p22p=306p−6p+2p22p\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.
306p−6p+2p22p=p−6p+2p22p\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.
5p−p+2p22p=5p−3−p\frac{5}{p} - \frac{ p +2 p^2}{2 p} = \frac{5}{p} -3 - p

STEP 6

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

STEP 7

Rearrange the inequality to isolate pp.
5p−p≥6\frac{5}{p} - p \geq6

STEP 8

Multiply through by pp to remove the fraction.
5−p2≥6p5 - p^2 \geq6p

STEP 9

Rearrange the inequality to form a quadratic inequality.
p2+6p−5≤p^2 +6p -5 \leq

STEP 10

Factor the quadratic inequality.
(p−)(p+5)≤0(p -)(p +5) \leq0

STEP 11

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

STEP 12

Test the intervals determined by the solutions to the inequality.For p<p <, (p−)(p+5)>0(p -)(p +5) >0.
For p>p >, (p−)(p+5)>0(p -)(p +5) >0.
So, the solution to the inequality is p>p >.
Therefore, the correct answer is p≥2p \geq2.

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