Solve each equation. See Example 1. 3/(x^2+x-2) - 1/(x^2-1) = 7/(2x^2+6x+4)

Verified step by step guidance

1. Factor each denominator: $x^2+x-2$ factors to $(x+2)(x-1)$, $x^2-1$ factors to $(x+1)(x-1)$, and $2x^2+6x+4$ factors to $2(x+2)(x+1)$.
2. Identify the least common denominator (LCD) for the fractions, which is $2(x+2)(x+1)(x-1)$.
3. Rewrite each fraction with the LCD as the new denominator: $\frac{3}{(x+2)(x-1)}$, $\frac{1}{(x+1)(x-1)}$, and $\frac{7}{2(x+2)(x+1)}$.
4. Multiply each term by the LCD to eliminate the denominators: $3\cdot \frac{2(x+1)}{2(x+2)(x+1)(x-1)}-1\cdot \frac{2(x+2)}{2(x+2)(x+1)(x-1)}=7\cdot \frac{(x-1)}{2(x+2)(x+1)(x-1)}$.
5. Simplify the resulting equation and solve for $x$ by combining like terms and isolating $x$.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of its factors. This is essential for simplifying expressions and solving equations. For example, the quadratic expression x^2 + x - 2 can be factored into (x - 1)(x + 2), which makes it easier to analyze and solve equations involving it.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. It allows us to combine fractions into a single expression. In the given equation, identifying a common denominator for the fractions on the left side will facilitate the simplification and solving of the equation.

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator and denominator. Solving these equations often requires finding values that do not make the denominator zero, as these values are excluded from the solution set. Understanding how to manipulate and solve rational equations is key to addressing the problem presented.