Solve rational equations

Learning Objectives Identify the domain of a rational equation and determine non-permissible values. Solve rational equations by multiplying by the Least Common Denominator (LCD). Solve rational equations that are proportions by using cross-multiplication. Algebraically identify and reject extraneous solutions. Verify the validity of a solution by substituting it back into the original equation. Connect the algebraic solution of f(x) = g(x) to the x-coordinate of the intersection point of the graphs y = f(x) and y = g(x). Model and solve real-world problems involving rates using rational equations. How can two pumps working together fill a tank faster than either one alone? 💧 Solving rational equations gives us the mathematical tools to answer this and many other rate-bas...

TermDefinitionExample Rational EquationAn equation that contains at least one rational expression, which is a fraction with a polynomial in the numerator and/or denominator.(x / (x - 3)) + (1 / (x + 1)) = 2 Domain (Non-Permissible Values)The set of all real numbers for which the equation is defined. For rational equations, this excludes any values of the variable that would make any denominator equal to zero.In the equation (5 / (x - 2)) = 10, the domain is all real numbers except x = 2. So, x ≠ 2. Least Common Denominator (LCD)The smallest polynomial that is a multiple of all the denominators present in the equation. It is found by factoring each denominator and taking the highest power of each unique factor.For the equation (1 / (x^2 - 4)) + (3 / (x + 2)) = 7, the denominators are (x-2)...

Method of Clearing Denominators Given an equation with rational expressions, multiply every term on both sides of the equation by the Least Common Denominator (LCD). This is the primary method for solving any rational equation. It effectively eliminates all fractions, transforming the rational equation into a polynomial equation (often linear or quadratic) that is much simpler to solve. Cross-Multiplication Property for Proportions If (P(x) / Q(x)) = (R(x) / S(x)), where Q(x) ≠ 0 and S(x) ≠ 0, then P(x) * S(x) = R(x) * Q(x). This is a shortcut for the 'Clearing Denominators' method that applies only when the equation consists of a single fraction equal to another single fraction. It simplifies the process by directly multiplying the numerator of each side by the de...