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Chapter 8: Problem 209

In the following exercises, subtract. $$ \frac{t}{t-6}-\frac{t-2}{t+6} $$

Short Answer

Expert verified
\frac{14t - 12}{(t-6)(t+6)}

Step by step solution

01

- Identify the common denominator

To subtract the fractions, we need a common denominator. The denominators we have are \(t-6\) and \(t+6\). The common denominator is the product of these two denominators: \( (t-6)(t+6) \).
02

- Rewrite each fraction with the common denominator

Rewrite \( \frac{t}{t-6} \) and \( \frac{t-2}{t+6} \) so both have the common denominator \( (t-6)(t+6) \). This becomes \(\frac{t(t+6)}{(t-6)(t+6)} - \frac{(t-2)(t-6)}{(t-6)(t+6)}\).
03

- Distribute terms in the numerators

Expand the numerators by distributing: \( \frac{t^2+6t}{(t-6)(t+6)} - \frac{t^2-8t+12}{(t-6)(t+6)} \).
04

- Subtract the numerators

Combine the numerators over the common denominator: \( \frac{(t^2 + 6t) - (t^2 - 8t + 12)}{(t-6)(t+6)} \). Simplify the numerator by combining like terms: \( t^2 + 6t - t^2 + 8t - 12 = 14t - 12 \).
05

- Simplify the final expression

The expression simplifies to: \( \frac{14t - 12}{(t-6)(t+6)} \). This is the result of the subtraction.

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Key Concepts

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

Common Denominator
When subtracting algebraic fractions, the first step is to identify a common denominator.
This denominator is a value that each individual fraction can be converted into.
In our exercise, the fractions are \(\frac{t}{t-6}\) and \(\frac{t-2}{t+6}\).
To find a common denominator, we multiply the two denominators together:
\((t-6)(t+6)\).
This ensures that both fractions share the same bottom part, making them easier to subtract.
Distributing Terms
Once a common denominator is established, rewrite each fraction to have this common denominator.
The exercise gives us \(\frac{t}{t-6}\) and \(\frac{t-2}{t+6}\).
We convert these to: \(\frac{t(t+6)}{(t-6)(t+6)}\) and \(\frac{(t-2)(t-6)}{(t-6)(t+6)}\).
Next, distribute the terms in the numerators.
This means multiplying each term inside the parentheses.
For \(\frac{t(t+6)}{(t-6)(t+6)}\), we get \(\frac{t^2 + 6t}{(t-6)(t+6)}\).
For \(\frac{(t-2)(t-6)}{(t-6)(t+6)}\), we get \(\frac{t^2 - 8t + 12}{(t-6)(t+6)}\).
Combining Like Terms
With the fractions rewritten to the same denominator, the next task is to subtract the numerators.
We now deal with \(\frac{t^2 + 6t}{(t-6)(t+6)}\) and \(\frac{t^2-8t+12}{(t-6)(t+6)}\).
Subtracting the numerators involves combining like terms:
\(\frac{(t^2 + 6t) - (t^2 - 8t + 12)}{(t-6)(t+6)}\).
Notice that when we open the parenthesis, the terms inside get visually more complex.
The simplified form of combining these is:
\(t^2 +6t - t^2 + 8t - 12 = 14t - 12\).
This step clarifies the simplified form of the numerator.
Simplifying Fractions
Finally, simplify the result into its neatest form.
Our current fraction is \(\frac{14t - 12}{(t-6)(t+6)}\).
Here, always check if you can factor out any common terms from the numerator.
In this exercise, 14 and 12 share a common factor of 2.
This results in \(\frac{2(7t - 6)}{(t-6)(t+6)}\).
If there's no further simplification possible, this becomes our final result.
Simplifying makes the fraction easier to understand and solve, especially in more complex algebra problems.

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