Problem 52
Question
For exercises \(35-86\), simplify. $$ \frac{v+1}{6 v-24}-\frac{v}{v-4} $$
Step-by-Step Solution
Verified Answer
Thus, the simplified form is \( \frac{-5v + 1}{6(v-4)}\).
1Step 1 - Factor the Denominators
First, factor any common terms in the denominators. For the expression \(\frac{v+1}{6v-24}-\frac{v}{v-4} \), notice that \(6v-24 = 6(v-4)\). The expression becomes: \(\frac{v+1}{6(v-4)}-\frac{v}{v-4}\).
2Step 2 - Find a Common Denominator
The common denominator for the two fractions is \(6(v-4)\), since \(v-4\) is a factor of both denominators.
3Step 3 - Rewrite Fractions with Common Denominator
Rewrite each fraction with the common denominator: \(\frac{v+1}{6(v-4)}-\frac{v \times 6}{(v-4) \times 6}\), simplifying to \(\frac{v+1}{6(v-4)}-\frac{6v}{6(v-4)}\).
4Step 4 - Combine the Fractions
Combine the fractions into a single expression: \(\frac{v+1-6v}{6(v-4)}\), which simplifies to \(\frac{v + 1 - 6v }{6(v-4)}\).
5Step 5 - Simplify the Numerator
Simplify the numerator by combining like terms: \( (v + 1 - 6v = -5v + 1) \), resulting in \(\frac{-5v + 1}{6(v-4)}\).
Key Concepts
Factoring DenominatorsFinding Common DenominatorsCombining FractionsLike Terms in Algebra
Factoring Denominators
Understanding how to factor denominators is crucial in simplifying algebraic fractions.
Factoring involves breaking down a polynomial or expression into the product of its simplest parts.
For instance, consider the exercise \(\frac{v+1}{6v-24}-\frac{v}{v-4} \).
Notice that the denominator 6v-24 can be factored as 6(v-4).
By factoring, we make the expressions easier to manage and find common denominators.
Applying this to our example, the fraction becomes: \(\frac{v+1}{6(v-4)}-\frac{v}{v-4} \).
This step helps to standardize the expression, making further simplification steps clearer. Remember, always look for common factors in the denominator that can simplify the problem.
Factoring involves breaking down a polynomial or expression into the product of its simplest parts.
For instance, consider the exercise \(\frac{v+1}{6v-24}-\frac{v}{v-4} \).
Notice that the denominator 6v-24 can be factored as 6(v-4).
By factoring, we make the expressions easier to manage and find common denominators.
Applying this to our example, the fraction becomes: \(\frac{v+1}{6(v-4)}-\frac{v}{v-4} \).
This step helps to standardize the expression, making further simplification steps clearer. Remember, always look for common factors in the denominator that can simplify the problem.
Finding Common Denominators
Finding a common denominator is essential when dealing with multiple fractions.
A common denominator allows us to combine fractions by giving them a shared base.
In the exercise, the denominators are 6(v-4) and v-4.
To combine these fractions, we need to identify a mutual denominator.
Here, 6(v-4) is the common denominator since v-4 is already a factor.
Rewriting each fraction with this common denominator transforms the original expression into \(\frac{v+1}{6(v-4)}-\frac{v \times 6}{(v-4) \times 6} \).
Simplifying to the common base allows us to handle these fractions more easily in subsequent steps.
A common denominator allows us to combine fractions by giving them a shared base.
In the exercise, the denominators are 6(v-4) and v-4.
To combine these fractions, we need to identify a mutual denominator.
Here, 6(v-4) is the common denominator since v-4 is already a factor.
Rewriting each fraction with this common denominator transforms the original expression into \(\frac{v+1}{6(v-4)}-\frac{v \times 6}{(v-4) \times 6} \).
Simplifying to the common base allows us to handle these fractions more easily in subsequent steps.
Combining Fractions
Combining fractions into one expression is simpler when they share a common denominator.
After finding the common denominator, we rewrite the fractions accordingly.
For our exercise, rewriting gives: \(\frac{v+1}{6(v-4)} - \frac{6v}{6(v-4)} \).
Once the fractions appear with the same denominator, we can combine them by subtracting the numerators directly.
This leads us to a single fraction: \(\frac{v+1-6v}{6(v-4)} \).
By combining the numerators over a shared base, we simplify the expression efficiently.
After finding the common denominator, we rewrite the fractions accordingly.
For our exercise, rewriting gives: \(\frac{v+1}{6(v-4)} - \frac{6v}{6(v-4)} \).
Once the fractions appear with the same denominator, we can combine them by subtracting the numerators directly.
This leads us to a single fraction: \(\frac{v+1-6v}{6(v-4)} \).
By combining the numerators over a shared base, we simplify the expression efficiently.
Like Terms in Algebra
In algebra, 'like terms' refer to terms that have the same variable raised to the same power.
Simplifying expressions often involves combining these like terms to make the expression more concise.
In the exercise, we combined the numerators: \ (v+1-6v) \ .
This step requires recognizing that both v and 6v are like terms.
Removing this commonality further simplifies the expression.
Simplify by rearranging \bin the expression leads us to \(\frac{-5v + 1}{6(v-4)} \).
Understanding the concept of like terms is vital in reducing expressions to their simplest form.
This final step ensures that our representation of the problem is as straightforward as possible.
Simplifying expressions often involves combining these like terms to make the expression more concise.
In the exercise, we combined the numerators: \ (v+1-6v) \ .
This step requires recognizing that both v and 6v are like terms.
Removing this commonality further simplifies the expression.
Simplify by rearranging \bin the expression leads us to \(\frac{-5v + 1}{6(v-4)} \).
Understanding the concept of like terms is vital in reducing expressions to their simplest form.
This final step ensures that our representation of the problem is as straightforward as possible.
Other exercises in this chapter
Problem 51
For exercises 39-82, simplify. $$ \frac{3 b}{8 d} \div \frac{3 b}{20 d} $$
View solution Problem 51
For exercises 1-66, simplify. $$ \frac{2 c^{2}-13 c-45}{2 c^{2}+13 c+20} $$
View solution Problem 52
For exercises 49-52, simplify. $$ \frac{m^{3}}{m^{2}+12 m+27}+\frac{27}{m^{2}+12 m+27} $$
View solution Problem 52
For exercises 39-82, simplify. $$ \frac{7 w}{9 p} \div \frac{7 w}{30 p} $$
View solution