Quick Answer
A rational equation is an algebraic equation containing at least one fraction with a variable in the denominator. On the Digital SAT, these typically appear in the Math section as medium-to-hard difficulty problems. Students encounter them approximately 2-4 times per test, often requiring solving for a specific variable or identifying extraneous solutions.
A rational equation is an equation that contains at least one rational expression, often appearing in the form $P(x)/Q(x) = R(x)/S(x)$ where $P, Q, R,$ and $S$ are polynomials. To solve these, one must eliminate the denominators, usually through cross-multiplication or multiplying by a common denominator.
Question: Solve for $x$: $\frac{3}{x+2} = \frac{1}{x}$. Solution: Cross-multiply to get $3x = 1(x+2)$. Simplify to $3x = x + 2$. Subtract $x$ from both sides: $2x = 2$. Divide by 2: $x = 1$. Since $x=1$ does not make either denominator zero, it is the valid solution.
Ignoring extraneous solutions: Students often solve the resulting polynomial but forget to check if the answer makes the original denominator zero.
Incorrect distribution: When multiplying by a common denominator, students frequently fail to distribute the term to every single part of the equation.
Sign errors during cross-multiplication: Mismanaging negative signs when multiplying binomials in the numerator is a frequent source of calculation errors.
Students targeting 750+ should know that the SAT frequently uses rational equations to test the concept of 'no solution' or 'infinitely many solutions' by creating scenarios where the variable cancels out or results in an undefined value.
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