Solve each equation. |3 - 2x | = |5 - 2x |

Verified step by step guidance

1. Recognize that you have an equation involving absolute values: \(|3 - 2x| = |5 - 2x|\).
2. Understand that the absolute value equation \(|A| = |B|\) implies two possible cases: either \(A = B\) or \(A = -B\).
3. Set up the first case: \(3 - 2x = 5 - 2x\).
4. Set up the second case: \(3 - 2x = -(5 - 2x)\).
5. Solve each case separately to find the possible values of \(x\).

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases for each equation.

Setting Up Cases

When solving equations involving absolute values, it is essential to set up separate cases based on the definitions of absolute value. For the equation |A| = |B|, we consider two scenarios: A = B and A = -B. This approach allows us to find all possible solutions by addressing both the positive and negative scenarios of the expressions involved.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. To solve these equations, we isolate the variable on one side of the equation through algebraic manipulation, such as adding, subtracting, multiplying, or dividing both sides. Mastery of solving linear equations is fundamental in algebra, as it forms the basis for more complex problem-solving.