Solve each equation. |3/ 2x - 1 | = 4

Verified step by step guidance

1. Start by understanding that the equation involves an absolute value: \(|\frac{3}{2x - 1}| = 4\). This means the expression inside the absolute value can be either positive or negative, leading to two separate equations.
2. Set up the first equation by removing the absolute value and setting the expression equal to 4: \(\frac{3}{2x - 1} = 4\).
3. Solve the first equation: Multiply both sides by \(2x - 1\) to eliminate the fraction, then solve for \(x\).
4. Set up the second equation by removing the absolute value and setting the expression equal to -4: \(\frac{3}{2x - 1} = -4\).
5. Solve the second equation: Again, multiply both sides by \(2x - 1\) to eliminate the fraction, then solve for \(x\).

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. In the context of equations, this means that |A| = B implies A = B or A = -B, which is crucial for solving equations involving absolute values.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves only linear terms and can be expressed in the form Ax + B = C, where A, B, and C are constants. Solving linear equations involves isolating the variable on one side of the equation. In the given problem, the absolute value equation leads to two separate linear equations that need to be solved.

Isolating the Variable

Isolating the variable is a fundamental technique in algebra used to solve equations. This involves manipulating the equation through addition, subtraction, multiplication, or division to get the variable alone on one side. In the context of the given absolute value equation, after breaking it into two cases, isolating x will help find the solutions for the original equation.