Problem 32 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{v^{2}}{2 v-14}-\frac{9 v-14}{2 v-14} $$

Step-by-Step Solution

Answer

\( \frac{v - 2}{2} \)

Step 1: Find a common denominator

Since both terms have the same denominator of \(2v-14\), we can combine them. The expression becomes \[ \frac{v^{2}}{2v-14} - \frac{9v-14}{2v-14} \text{ which can be written as } \frac{v^{2} - (9v - 14)}{2v-14}.\] Remember to distribute the negative sign in the numerator.

Step 2: Simplify the numerator

Distribute the negative sign and combine like terms:\[ v^{2} - (9v - 14) = v^{2} - 9v + 14.\]

Step 3: Rewrite the expression in simplified form

Now the expression is: \[ \frac{v^{2} - 9v + 14}{2v - 14}.\]

Step 4: Check for further simplification

Factor the numerator if possible. We have \( v^{2} - 9v + 14 \) which factors to \( (v - 7)(v - 2). \) So, the expression becomes \[ \frac{(v - 7)(v - 2)}{2(v-7)},\] because \(2v - 14 = 2(v-7).\) Therefore, cancel the \(v-7\) terms, leading to \[ \frac{v - 2}{2}.\]

Step 5: Final simplified form

The simplified expression is \[ \frac{v - 2}{2}.\]

Key Concepts

Common Denominator

In rational expressions, a common denominator allows us to combine fractions easily. The common denominator for \(\frac{v^{2}}{2v-14}\) and \(\frac{9v-14}{2v-14}\) is \2v-14\. Since both fractions already share this denominator, we can combine them.

Think of the denominator like the base for the fractions. By matching these bases, we can directly subtract the numerators, making the process simpler.

When the denominators are the same, subtraction of rational expressions becomes:

\[\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\]

In our example, this process leads to rewriting the expression as: \ \frac{v^{2} - (9v - 14)}{2v - 14}\.

Distribute Negative Sign

Distributing the negative sign is a key step when simplifying numerators. In our exercise, we need to subtract \(9v - 14\) from \v^{2}\. To do this correctly, we must distribute the negative sign across each term inside the parentheses:

\[v^{2} - (9v - 14) = v^{2} - 9v + 14\]

This distributes the negative sign to both terms inside the parenthesis:

- Convert \- (9v) \ to \-9v\.
- Convert \- (-14)\ to \+14\.

Remembering to apply the negative to each term prevents mistakes and ensures accurate simplification.

After this, we combine like terms to get the simplified numerator.

Factorization

Factorization is the process of breaking down an expression into simpler 'factors' or components. In this exercise, we factor the quadratic expression \v^{2} - 9v + 14\ to see if any simplification or cancellation is possible:

1. Identify terms that multiply to give the constant term (14) and add to give the middle coefficient (-9).
2. This gives us factors (v - 7) and (v - 2):

\[v^{2} - 9v + 14 = (v - 7)(v - 2)\]

Next, we notice the denominator \2v - 14\ can be rewritten using factorization as well:

\[2v - 14 = 2(v - 7)\]

Now our expression is:

\[ \frac{(v - 7)(v - 2)}{2(v - 7)} \]

Finally, we cancel out the common factor (v - 7) from numerator and denominator, resulting in the simplified form:

\[ \frac{v - 2}{2} \]