The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4/(x² + 3x - 10) - 1/(x² + x - 6) = 3/(x² - x - 12)

Verified step by step guidance

1. Factorize the denominators of each term in the equation. For example, factorize \(x^2 + 3x - 10\), \(x^2 + x - 6\), and \(x^2 - x - 12\) into their respective factored forms.
2. Identify the least common denominator (LCD) of the equation by combining all unique factors from the denominators.
3. Multiply through the entire equation by the LCD to eliminate the denominators. This will leave you with a polynomial equation.
4. Simplify the resulting polynomial equation by combining like terms and rearranging it into standard form, \(ax^2 + bx + c = 0\).
5. Solve the simplified polynomial equation using appropriate methods such as factoring, completing the square, or the quadratic formula. Check the solutions in the original equation to ensure they do not make any denominator zero, and classify the equation as an identity, conditional, or inconsistent based on the results.

Key Concepts

Types of Equations

In algebra, equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications helps in determining the nature of the solution set for a given equation.

Factoring Polynomials

Factoring polynomials is a crucial skill in solving rational equations. It involves expressing a polynomial as a product of its factors, which can simplify the equation and make it easier to solve. For example, the expressions in the denominators of the given equation can be factored to identify common terms and simplify the overall equation.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. When solving equations involving rational expressions, it is essential to find a common denominator to combine the fractions effectively. Additionally, one must be cautious of restrictions on the variable that can arise from the denominators, as these can lead to undefined values.