Solve each equation. | 6x + 1/ x - 1 | = 3

Verified step by step guidance

1. Identify the absolute value equation: \(|\frac{6x + 1}{x - 1}| = 3\).
2. Set up two separate equations to account for the absolute value: \(\frac{6x + 1}{x - 1} = 3\) and \(\frac{6x + 1}{x - 1} = -3\).
3. Solve the first equation: \(\frac{6x + 1}{x - 1} = 3\). Multiply both sides by \(x - 1\) to eliminate the fraction, resulting in \(6x + 1 = 3(x - 1)\).
4. Solve the second equation: \(\frac{6x + 1}{x - 1} = -3\). Similarly, multiply both sides by \(x - 1\) to eliminate the fraction, resulting in \(6x + 1 = -3(x - 1)\).
5. Solve each resulting linear equation for \(x\) and check for any extraneous solutions by substituting back into the original equation.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, disregarding its sign. In the equation |6x + 1/(x - 1)| = 3, the absolute value indicates that the expression inside can equal either 3 or -3. Understanding how to manipulate absolute values is crucial for solving equations that involve them.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In this equation, 1/(x - 1) is a rational expression, and it is important to consider its domain, as it cannot equal zero. This understanding helps in identifying restrictions on the variable x when solving the equation.

Equating Expressions

When solving equations, particularly those involving absolute values, it is essential to set up separate equations based on the possible cases. For |A| = B, we create A = B and A = -B. This method allows us to find all potential solutions, which we can then verify to ensure they satisfy the original equation.