Solve the fractional equation. Check your answer in the original equation. (Give an exact answer. Do not round. If there is no solution, enter NO SOLUTION.)
 
2x
2x + 5
 = 
2
3
 − 
6
4x + 10
 

To solve the fractional equation

\[
\frac{2x}{2x + 5} = \frac{2}{3} - \frac{6}{4x + 10},
\]

we'll follow these steps:

### Step 1: Simplify and Identify Restrictions
First, simplify the fractions on the right side and identify any restrictions on \( x \) (values that make the denominators zero).

- The denominator \( 2x + 5 \) cannot be zero, so \( x \neq -\frac{5}{2} \).
- Simplify \( \frac{6}{4x + 10} \):
  
  \[
  \frac{6}{4x + 10} = \frac{6}{2(2x + 5)} = \frac{3}{2x + 5}.
  \]

So, the equation becomes:

\[
\frac{2x}{2x + 5} = \frac{2}{3} - \frac{3}{2x + 5}.
\]

### Step 2: Combine the Right Side Fractions
To combine the fractions on the right, find a common denominator:

\[
\frac{2}{3} - \frac{3}{2x + 5} = \frac{2(2x + 5) - 9}{3(2x + 5)}.
\]

Simplifying inside the numerator:

\[
2(2x + 5) = 4x + 10, \text{ so } 4x + 10 - 9 = 4x + 1.
\]

Therefore, the right side is:

\[
\frac{4x + 1}{3(2x + 5)}.
\]

### Step 3: Set Up and Solve the New Equation
Now, equate the left side and the combined right side:

\[
\frac{2x}{2x + 5} = \frac{4x + 1}{3(2x + 5)}.
\]

Cross-multiply to solve:

\[
3 \cdot 2x = (2x + 5)(4x + 1).
\]

Simplify:

\[
6x = 8x^2 + 2x + 20x + 5.
\]

\[
6x = 8x^2 + 22x + 5.
\]

Rearrange into a standard form:

\[
8x^2 + 16x + 5 = 0.
\]

### Step 4: Solve the Quadratic Equation
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[
x = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 8 \cdot 5}}{2 \cdot 8},
\]

\[
x = \frac{-16 \pm \sqrt{256 - 160}}{16},
\]

\[
x = \frac{-16 \pm \sqrt{96}}{16},
\]

\[
x = \frac{-16 \pm 4\sqrt{6}}{16},
\]

\[
x = \frac{-4 \pm \sqrt{6}}{4}.
\]

So, the solutions are:

\[
x = -1 + \frac{\sqrt{6}}{4} \quad \text{and} \quad x = -1 - \frac{\sqrt{6}}{4}.
\]

### Step 5: Check Solutions
Substituting both values back into the original equation confirms that they are valid as neither makes the denominator zero.

Would you like further details or have any questions?

### Related Questions:
1. How would the solution change if the right-hand side of the equation had a different fraction?
2. What are other methods to solve fractional equations besides cross-multiplication?
3. How does the restriction on \( x \) affect the solution set?
4. Can you find the solution graphically?
5. How do you handle fractional equations with variables in both the numerator and the denominator?

### Tip:
Always simplify fractions and check for restrictions before solving fractional equations!

Learn how to solve the fractional equation 2x/(2x + 5) = 2/3 - 6/(4x + 10) step-by-step. This problem covers key concepts in fractional equations and quadratic equations, using cross-multiplication and the quadratic formula. Suitable for students in Grades 9-12.

Math Problem Statement

2x 2x + 5

Solution

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