Solve the rational equation. If there is more than one correct answer, enter a comma separated list. If there is no answer, enter “NONE”.
v−12/v2−9v+18=4/v−3−5/v−6

We are asked to solve the rational equation:

\[
\frac{v-12}{v^2 - 9v + 18} = \frac{4}{v - 3} - \frac{5}{v - 6}
\]

### Step 1: Simplify the equation

First, factor the quadratic expression in the denominator on the left side:

\[
v^2 - 9v + 18 = (v - 3)(v - 6)
\]

So, the equation becomes:

\[
\frac{v-12}{(v-3)(v-6)} = \frac{4}{v - 3} - \frac{5}{v - 6}
\]

### Step 2: Combine the fractions on the right side

To combine the fractions on the right, we need a common denominator, which is \((v-3)(v-6)\):

\[
\frac{4}{v - 3} = \frac{4(v - 6)}{(v - 3)(v - 6)} = \frac{4v - 24}{(v - 3)(v - 6)}
\]

\[
\frac{5}{v - 6} = \frac{5(v - 3)}{(v - 3)(v - 6)} = \frac{5v - 15}{(v - 3)(v - 6)}
\]

Subtract the fractions:

\[
\frac{4v - 24}{(v - 3)(v - 6)} - \frac{5v - 15}{(v - 3)(v - 6)} = \frac{4v - 24 - (5v - 15)}{(v - 3)(v - 6)}
\]

Simplify the numerator:

\[
4v - 24 - 5v + 15 = -v - 9
\]

So the right side of the equation becomes:

\[
\frac{-v - 9}{(v - 3)(v - 6)}
\]

### Step 3: Equate the numerators

Since the denominators on both sides are equal, we can equate the numerators:

\[
v - 12 = -v - 9
\]

### Step 4: Solve the equation

Add \(v\) to both sides:

\[
2v - 12 = -9
\]

Add 12 to both sides:

\[
2v = 3
\]

Divide by 2:

\[
v = \frac{3}{2}
\]

### Step 5: Check for extraneous solutions

We need to make sure \(v = \frac{3}{2}\) does not make any denominator zero. The denominators are \(v - 3\) and \(v - 6\). Substituting \(v = \frac{3}{2}\) into these expressions:

\[
v - 3 = \frac{3}{2} - 3 = \frac{3}{2} - \frac{6}{2} = \frac{-3}{2} \quad (\text{not zero})
\]
\[
v - 6 = \frac{3}{2} - 6 = \frac{3}{2} - \frac{12}{2} = \frac{-9}{2} \quad (\text{not zero})
\]

Since neither denominator is zero, \(v = \frac{3}{2}\) is a valid solution.

### Final Answer:
\[
v = \frac{3}{2}
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you factor quadratic expressions like \(v^2 - 9v + 18\)?
2. What are extraneous solutions and how do they arise in rational equations?
3. Can you always equate the numerators when the denominators are equal in rational equations?
4. How do you find a common denominator for adding or subtracting rational expressions?
5. What happens if a solution makes a denominator zero?

### Tip:
When solving rational equations, always check your solutions against the original denominators to ensure they don't cause division by zero.

Learn how to solve the rational equation (v-12)/(v^2 - 9v + 18) = 4/(v-3) - 5/(v-6) step-by-step with a detailed solution. This problem involves rational equations, quadratic equations, and fraction operations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation