Solve each equation. |x + 1 | = |1 -3x|

Verified step by step guidance

1. Recognize that the equation involves absolute values: \(|x + 1| = |1 - 3x|\). To solve it, 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 \(A = B\) or \(A = -B\). So, write the two cases: \(x + 1 = 1 - 3x\) and \(x + 1 = -(1 - 3x)\).
3. Solve the first equation \(x + 1 = 1 - 3x\) by isolating \(x\). Subtract 1 from both sides and then add \$3x$ to both sides to combine like terms.
4. Solve the second equation \(x + 1 = -1 + 3x\) by distributing the negative sign and then isolating \(x\). Move all \(x\) terms to one side and constants to the other.
5. Check each solution by substituting back into the original equation to ensure both sides are equal, confirming the solutions are valid.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For any expression |A| = |B|, it means either A = B or A = -B, which is essential for setting up equations to solve.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side using inverse operations like addition, subtraction, multiplication, or division. This skill is necessary after splitting the absolute value equation into two cases to find possible solutions.

Checking Solutions in Absolute Value Equations

Not all solutions obtained from splitting absolute value equations are valid. It's important to substitute solutions back into the original equation to verify they satisfy the absolute value condition, ensuring no extraneous solutions are included.