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

Verified step by step guidance

1. Recognize that the equation involves an absolute value expression: \(\left| \frac{6x + 1}{x - 1} \right| = 3\). The absolute value means the expression inside can be either 3 or -3.
2. Set up two separate equations to remove the absolute value: \(\frac{6x + 1}{x - 1} = 3\) and \(\frac{6x + 1}{x - 1} = -3\).
3. Solve the first equation \(\frac{6x + 1}{x - 1} = 3\) by multiplying both sides by \((x - 1)\) to clear the denominator: \$6x + 1 = 3(x - 1)$.
4. Expand and simplify the first equation: \$6x + 1 = 3x - 3\(, then isolate \)x$ by moving terms to one side.
5. Repeat the process for the second equation \(\frac{6x + 1}{x - 1} = -3\): multiply both sides by \((x - 1)\), expand, simplify, and solve for \(x\). Remember to check for any restrictions such as \(x \neq 1\) because the denominator cannot be zero.

Key Concepts

Absolute Value Equations

An absolute value equation involves expressions within absolute value bars, which represent the distance from zero on the number line. To solve, set the expression inside equal to both the positive and negative value of the number outside, creating two separate equations to solve.

Rational Expressions

A rational expression is a fraction where the numerator and/or denominator are polynomials. When solving equations involving rational expressions, it is important to consider restrictions on the variable that make the denominator zero, as these values are excluded from the solution.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side using inverse operations such as addition, subtraction, multiplication, and division. After setting up equations from the absolute value definition, solve each linear equation carefully to find possible solutions.