- Example 1 Solution:
- Start with the equation: $\frac{10}{2x - 3} = 2$.
- Multiply both sides by $(2x - 3)$ to eliminate the denominator: $10 = 2(2x - 3)$.
- Distribute the 2 on the right side: $10 = 4x - 6$.
- Add 6 to both sides: $16 = 4x$.
- Divide both sides by 4: $x = 4$.
- Check for extraneous solutions: Substitute $x = 4$ into the original denominator $2x - 3$. $2(4) - 3 = 5 \neq 0$, so $x = 4$ is valid.
- Example 2 Solution:
- Start with the equation: $\frac{x - 5}{x^2 - 2x - 8} = \frac{5x - 1}{x^2 - 4}$.
- Factor the denominators: $x^2 - 2x - 8 = (x - 4)(x + 2)$ and $x^2 - 4 = (x - 2)(x + 2)$.
- The equation becomes: $\frac{x - 5}{(x - 4)(x + 2)} = \frac{5x - 1}{(x - 2)(x + 2)}$.
- Multiply both sides by the least common denominator, $(x - 4)(x + 2)(x - 2)$, to clear fractions: $(x - 5)(x - 2) = (5x - 1)(x - 4)$.
- Expand both sides:
- Left: $(x - 5)(x - 2) = x^2 - 2x - 5x + 10 = x^2 - 7x + 10$.
- Right: $(5x - 1)(x - 4) = 5x^2 - 20x - x + 4 = 5x^2 - 21x + 4$.
- Bring all terms to one side: $x^2 - 7x + 10 - (5x^2 - 21x + 4) = 0$.
- Simplify: $x^2 - 7x + 10 - 5x^2 + 21x - 4 = 0 \Rightarrow -4x^2 + 14x + 6 = 0$.
- Multiply by -1 to simplify: $4x^2 - 14x - 6 = 0$.
- Divide by 2: $2x^2 - 7x - 3 = 0$.
- Use the quadratic formula: $x = \frac{7 \pm \sqrt{(-7)^2 - 4(2)(-3)}}{2(2)} = \frac{7 \pm \sqrt{49 + 24}}{4} = \frac{7 \pm \sqrt{73}}{4}$.
- Check for extraneous solutions: The original denominators are zero when $x = 4$, $x = -2$, or $x = 2$. The solutions $\frac{7 \pm \sqrt{73}}{4}$ are approximately $x \approx 3.89$ and $x \approx -0.39$, neither of which are excluded values, so both are valid.
Parent Tip: Review the logic above to help your child master the concept of rational inequalities worksheet.
