Add or subtract as indicated. [(2x-7)/(x^2-9)]-[(x-10)/(x^2-9)]

Verified step by step guidance

1. Step 1: Recognize that the denominators in both fractions are the same, \(x^2 - 9\). This means we can combine the numerators directly by subtraction.
2. Step 2: Write the combined fraction as \(\frac{(2x - 7) - (x - 10)}{x^2 - 9}\). Be careful to distribute the negative sign across the second numerator.
3. Step 3: Simplify the numerator by combining like terms. Expand \((2x - 7) - (x - 10)\) to \(2x - 7 - x + 10\), which simplifies to \(x + 3\).
4. Step 4: Rewrite the fraction as \(\frac{x + 3}{x^2 - 9}\).
5. Step 5: Factor the denominator \(x^2 - 9\) using the difference of squares formula: \(x^2 - 9 = (x - 3)(x + 3)\). Check if the numerator \(x + 3\) can cancel with a factor in the denominator.

Key Concepts

Common Denominator

In order to add or subtract fractions, they must have a common denominator. In this case, both fractions share the denominator (x^2 - 9), which simplifies the process. When the denominators are the same, you can combine the numerators directly while keeping the common denominator intact.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. In this problem, recognizing that x^2 - 9 can be factored into (x - 3)(x + 3) is crucial for simplifying the expression and understanding its behavior.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing the expression to its simplest form by canceling out common factors in the numerator and denominator. After combining the fractions, it is important to simplify the resulting expression to make it easier to interpret and work with, ensuring that any restrictions on the variable are also noted.