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

Short Answer

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

Step by step solution

01

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: $$(n-2)(n+1)$$.
02

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: $$0=\frac{2(n+1)}{(n-2)(n+1)}-\frac{6(n-2)}{(n-2)(n+1)}$$.
03

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: $$0=\frac{-4n+14}{(n-2)(n+1)} $$
04

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: $$n=\frac{7}{2}$$
05

Final Answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions
Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). Understanding fractions is crucial in many areas of math, especially when dealing with equations like \[0=\frac{2}{n-2}-\frac{6}{n+1}\]
Fractions are handy because they allow us to express divisions and ratios, solving problems that involve parts of a whole.
To work with fractions efficiently, you must grasp these concepts:
  • Numerator and Denominator: The numerator tells us the number of equal parts we have, while the denominator indicates how many of those parts make up a whole.
  • Operations with Fractions: Adding, subtracting, multiplying, and dividing fractions involves working with these parts, often by finding common denominators or cross-multiplying.
Common Denominator
Finding a common denominator is a vital step when adding or subtracting fractions. It allows you to combine fractions by ensuring they share the same base.Consider the expression: \[0=\frac{2}{n-2}-\frac{6}{n+1}\]
Here, the different denominators make it hard to subtract them directly. Instead, we find a common denominator, which is often the least common multiple (LCM) of the denominators. In our exercise, the denominators are \( (n-2) \) and \( (n+1) \).
  • The common denominator is \( (n-2)(n+1) \), which combines both bases.
  • This enables us to express each fraction with the same denominator, making them easier to handle.
By expressing fractions in terms of a common denominator, you simplify the process of combining them, which is a fundamental skill in algebra.
Simplifying Expressions
Simplifying expressions is about making them easier to work with while maintaining their value. In algebra, this often involves distributing, combining like terms, and reducing fractions.
For our equation, once fractions are written with a common denominator:\[0=\frac{2(n+1)-6(n-2)}{(n-2)(n+1)}\]
You simplify it by:
  • Distributing coefficients: Multiply to simplify the terms \( 2(n+1) \) and \( -6(n-2) \).
  • Combining like terms: Collect terms with similar variables or powers.
  • You get: \(\frac{-4n+14}{(n-2)(n+1)}\)
Simplifying makes it clearer and prepares the expression for further problem-solving, such as finding unknowns.
Solving Equations
Solving equations involves finding the value of unknown variables that make the equation true. It's like piecing together a puzzle by deducing the right number to fit into the equation.
In our example, when we simplify to:\[0=\frac{-4n+14}{(n-2)(n+1)}\]
We focus on the numerator because a fraction is zero only when its numerator is zero.
  • Set \(-4n+14=0\) to isolate \(n\).
  • Solve linear equations: Rearrange to solve, starting with subtraction and division.
  • The steps look like: \(-4n = -14\) then divide by \(-4\) to find \(n=\frac{7}{2}\).
By understanding each part methodically, you can solve complex equations by breaking them down into simple, manageable steps.

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