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

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|\frac{3}{2}x - 1| = 4\). Recall that \(|A| = B\) means \(A = B\) or \(A = -B\).
2. Set up two separate equations based on the definition of absolute value: \(\frac{3}{2}x - 1 = 4\) and \(\frac{3}{2}x - 1 = -4\).
3. Solve the first equation \(\frac{3}{2}x - 1 = 4\) by adding 1 to both sides: \(\frac{3}{2}x = 5\), then multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\), to isolate \(x\).
4. Solve the second equation \(\frac{3}{2}x - 1 = -4\) by adding 1 to both sides: \(\frac{3}{2}x = -3\), then multiply both sides by \(\frac{2}{3}\) to isolate \(x\).
5. Write the two solutions for \(x\) obtained from the two equations. These are the solutions to the original absolute value equation.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For an equation |A| = B, where B is positive, the expression inside the absolute value, A, can equal either B or -B, leading to two possible equations to solve.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, or division. This process helps find the value(s) of the variable that satisfy the equation.

Handling Fractions in Equations

When equations contain fractions, it is often helpful to clear denominators by multiplying both sides by the least common denominator. This simplifies the equation and makes it easier to isolate the variable and solve.