Question

If \(0=\frac{2}{n-2}-\frac{6}{n+1}\), what is the value of \(n\) ?

Short answer

The value of \(n\) that satisfies the given equation is \(n = \frac{7}{2}\).

Step-by-step solution

Find the common denominator

To find the common denominator, we need to find the least common multiple (LCM) of the denominators \((n-2)\) and \((n+1)\). In this case, the LCM is simply the product of the two denominators, which is: $$(n-2)(n+1)$$.

Express the fractions using the common denominator

Now, we need to express both fractions using the common denominator we found in step 1. To do this, we will multiply the numerator and denominator of each fraction by the appropriate factor. For the first fraction, the factor is \((n+1)\), and for the second fraction, the factor is \((n-2)\). This gives us the following equation: $$0=\frac{2(n+1)}{(n-2)(n+1)}-\frac{6(n-2)}{(n-2)(n+1)}$$.

Simplify the equation

Since both fractions in the equation already have the same denominator, we can combine them. This will give us the following equation: $$0=\frac{2(n+1)-6(n-2)}{(n-2)(n+1)}$$ Now, we will simplify the numerator by distributing and combining like terms: $$0=\frac{2n+2-6n+12}{(n-2)(n+1)} $$ This simplifies to: $$0=\frac{-4n+14}{(n-2)(n+1)} $$

Solve for \(n\)

Since we have a single fraction equal to zero, we can isolate \(n\) by setting the numerator of the fraction equal to zero (because a fraction is only equal to zero if its numerator is zero): $$-4n+14=0$$ To solve for \(n\), we'll first subtract 14 from both sides of the equation: $$-4n=-14$$ Now, we'll divide by -4: $$n=\frac{-14}{-4}$$ This simplifies to: $$n=\frac{7}{2}$$

Final Answer

The value of \(n\) that satisfies the given equation is \(n = \frac{7}{2}\).