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

Verified step by step guidance

1. Identify the equation given: \(\left| \frac{3}{2}x - 1 \right| = 4\). The absolute value expression means the quantity inside the absolute value can be either positive or negative but its distance from zero is 4.
2. Set up two separate equations to remove the absolute value: one where the inside expression equals 4, and one where it equals -4. So, write: \(\frac{3}{2}x - 1 = 4\) and \(\frac{3}{2}x - 1 = -4\).
3. Solve the first equation \(\frac{3}{2}x - 1 = 4\) by isolating \(x\). Add 1 to both sides to get \(\frac{3}{2}x = 5\), then multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\), to find \(x\).
4. Solve the second equation \(\frac{3}{2}x - 1 = -4\) similarly. Add 1 to both sides to get \(\frac{3}{2}x = -3\), then multiply both sides by \(\frac{2}{3}\) to find \(x\).
5. Write the two solutions for \(x\) obtained from the two equations. These are the values of \(x\) that satisfy the original absolute value equation.

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 the absolute value equal to both the positive and negative values of the number on the other side of the equation.

Isolating the Variable

Before solving an equation, isolate the variable term by performing inverse operations such as addition, subtraction, multiplication, or division. This simplifies the equation and makes it easier to solve for the variable.

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. After isolating the variable, solve the resulting linear equations by performing arithmetic operations to find the variable's value.