Solve each equation or inequality. |6 - 2x | + 1 = 3

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 1 from both sides of the equation to get: \(|6 - 2x| = 3 - 1\).
2. Simplify the right side to have: \(|6 - 2x| = 2\).
3. Recall that if \(|A| = B\), then \(A = B\) or \(A = -B\). Apply this to get two separate equations: \$6 - 2x = 2\( and \)6 - 2x = -2$.
4. Solve each equation separately. For \$6 - 2x = 2\(, subtract 6 from both sides and then divide by -2 to isolate \)x\(. For \)6 - 2x = -2$, do the same: subtract 6 and divide by -2.
5. Write the solutions for \(x\) from both equations. These values 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 expression |A| = B, it means A = B or A = -B, provided B ≥ 0. Understanding this helps in splitting the equation into two separate cases to solve.

Solving Linear Equations

Linear equations involve variables raised only to the first power and can be solved by isolating the variable using inverse operations like addition, subtraction, multiplication, and division. After splitting the absolute value equation, each resulting linear equation is solved separately.

Checking for Extraneous Solutions

When solving absolute value equations, some solutions may not satisfy the original equation due to the definition of absolute value. It is important to substitute solutions back into the original equation to verify their validity and discard any extraneous solutions.