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

Verified step by step guidance

1. Recognize that the equation involves absolute values: \(|3 - 2x| = |5 - 2x|\). To solve this, consider the definition of absolute value, which means the expressions inside the absolute values can be equal or opposites.
2. Set up two separate equations based on the property \(|A| = |B|\) implies either \(A = B\) or \(A = -B\). So, write the two cases: \$3 - 2x = 5 - 2x\( and \)3 - 2x = -(5 - 2x)$.
3. Solve the first equation \$3 - 2x = 5 - 2x\(. Simplify both sides and isolate \)x$ to find any possible solutions.
4. Solve the second equation \$3 - 2x = -5 + 2x\(. Simplify and solve for \)x\( by combining like terms and isolating \)x$.
5. Check the solutions obtained by substituting them back into the original equation to verify they satisfy the absolute value equality.

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 any expression |A|, it equals A if A is non-negative, and -A if A is negative. Understanding this helps in setting up equations when absolute values are involved.

Solving Absolute Value Equations

To solve equations involving absolute values, split the equation into two cases: one where the expressions inside the absolute values are equal, and one where they are opposites. This approach allows you to remove the absolute value signs and solve linear equations.

Linear Equation Solving

After removing absolute values, you solve the resulting linear equations by isolating the variable. This involves using inverse operations like addition, subtraction, multiplication, or division to find the value of the variable that satisfies the equation.