The problem involves solving the equation:
\[
\frac{1}{2x + 5} = \frac{x}{11x + 8}
\]
Step-by-Step Solution:
#### 1.
Eliminate the fractions by cross-multiplying:
To eliminate the fractions, we multiply both sides of the equation by the denominators \( (2x + 5) \) and \( (11x + 8) \):
\[
1 \cdot (11x + 8) = x \cdot (2x + 5)
\]
This simplifies to:
\[
11x + 8 = x(2x + 5)
\]
#### 2.
Expand the right-hand side:
Distribute \( x \) on the right-hand side:
\[
11x + 8 = 2x^2 + 5x
\]
#### 3.
Rearrange the equation into standard quadratic form:
To solve for \( x \), we need to bring all terms to one side of the equation. Subtract \( 11x \) and \( 8 \) from both sides:
\[
0 = 2x^2 + 5x - 11x - 8
\]
Simplify the equation:
\[
0 = 2x^2 - 6x - 8
\]
Rearrange it to standard quadratic form:
\[
2x^2 - 6x - 8 = 0
\]
#### 4.
Simplify the quadratic equation (if possible):
The quadratic equation \( 2x^2 - 6x - 8 = 0 \) can be simplified by dividing every term by 2:
\[
x^2 - 3x - 4 = 0
\]
#### 5.
Factor the quadratic equation:
We need to factor \( x^2 - 3x - 4 \). Look for two numbers that multiply to \(-4\) (the constant term) and add up to \(-3\) (the coefficient of \( x \)).
The numbers are \(-4\) and \(+1\) because:
\[
(-4) \cdot (+1) = -4 \quad \text{and} \quad (-4) + (+1) = -3
\]
Thus, we can factor the quadratic as:
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
#### 6.
Solve for \( x \) using the factored form:
Set each factor equal to zero:
\[
x - 4 = 0 \quad \text{or} \quad x + 1 = 0
\]
Solve these equations:
\[
x = 4 \quad \text{or} \quad x = -1
\]
#### 7.
Check for extraneous solutions:
Since the original equation involves fractions, we must ensure that the solutions do not make any denominator zero. The denominators are \( 2x + 5 \) and \( 11x + 8 \).
- For \( x = 4 \):
\[
2(4) + 5 = 8 + 5 = 13 \neq 0
\]
\[
11(4) + 8 = 44 + 8 = 52 \neq 0
\]
So, \( x = 4 \) is valid.
- For \( x = -1 \):
\[
2(-1) + 5 = -2 + 5 = 3 \neq 0
\]
\[
11(-1) + 8 = -11 + 8 = -3 \neq 0
\]
So, \( x = -1 \) is also valid.
#### Final Answer:
Both solutions are valid, so the solutions to the equation are:
\[
\boxed{4, -1}
\]
Parent Tip: Review the logic above to help your child master the concept of solving rational equations by cross multiplying worksheet.
