Questions: Solve the rational equation. Express numbers as integers or simplified fractions. p/(p-2) + 2/3 = 3/(p-2) The solution set is .

$Solve the rational equation. Express numbers as integers or simplified fractions. p/(p-2) + 2/3 = 3/(p-2) The solution set is .$

Transcript text: Solve the rational equation. Express numbers as integers or simplified fractions. \[ \frac{p}{p-2}+\frac{2}{3}=\frac{3}{p-2} \] The solution set is $\square$.

Solution

Solution Steps

Step 1: Find the least common denominator

The least common denominator (LCD) of the fractions is $3(p-2)$.

Step 2: Multiply both sides of the equation by the LCD

Multiplying both sides of the equation by $3(p-2)$, we get $3(p-2) \cdot \frac{p}{p-2} + 3(p-2) \cdot \frac{2}{3} = 3(p-2) \cdot \frac{3}{p-2}$ $3p + 2(p-2) = 9$

Step 3: Simplify and solve for p

Simplifying the equation gives $3p + 2p - 4 = 9$ $5p - 4 = 9$ $5p = 13$ $p = \frac{13}{5}$

Step 4: Check for extraneous solutions

Since $p = \frac{13}{5}$ does not make the denominator zero, it is not an extraneous solution.