Solve each equation. 10/4x-4 = 1 /1-x

Verified step by step guidance

1. Start by rewriting the equation clearly: \(\frac{10}{4x - 4} = \frac{1}{1 - x}\).
2. Identify the denominators on both sides: \$4x - 4\( and \)1 - x\(. Notice that \)4x - 4\( can be factored as \)4(x - 1)$.
3. To eliminate the fractions, cross-multiply the equation: \$10 \times (1 - x) = 1 \times (4x - 4)$.
4. Expand both sides: \$10 - 10x = 4x - 4$.
5. Collect like terms to isolate \(x\) on one side: add \$10x\( to both sides and add \)4\( to both sides, then solve for \)x$.

Key Concepts

Solving Rational Equations

Rational equations involve fractions with variables in the denominator. To solve them, find a common denominator or cross-multiply to eliminate fractions, then solve the resulting equation. Always check for values that make denominators zero, as these are excluded from the solution.

Cross-Multiplication

Cross-multiplication is a method used to solve equations where two fractions are set equal. Multiply the numerator of each fraction by the denominator of the other, creating a simpler equation without fractions. This technique is especially useful for equations like a/b = c/d.

Domain Restrictions in Rational Expressions

Variables in denominators cannot take values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is crucial before solving to avoid invalid solutions. After solving, verify that solutions do not violate these domain restrictions.