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

Verified step by step guidance

1. Recognize that the equation involves absolute values: \(|x + 1| = |1 - 3x|\). This means we need to consider the different cases where the expressions inside the absolute values can be positive or negative.
2. Set up the first case where both expressions inside the absolute values are equal: \(x + 1 = 1 - 3x\). Solve this equation for \(x\).
3. Set up the second case where the expression inside the first absolute value is equal to the negative of the expression inside the second absolute value: \(x + 1 = -(1 - 3x)\). Solve this equation for \(x\).
4. Check the solutions obtained from both cases to ensure they satisfy the original equation, as sometimes extraneous solutions can arise when dealing with absolute values.
5. Conclude with the valid solutions that satisfy the original equation \(|x + 1| = |1 - 3x|\).

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. This concept is crucial for solving equations involving absolute values, as it leads to the creation of two separate cases to consider.

Setting Up Cases

When solving equations that involve absolute values, it is essential to set up cases based on the definitions of absolute value. For the equation |A| = |B|, we can derive two scenarios: A = B and A = -B. This approach allows us to explore all possible solutions that satisfy the original equation.

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. Understanding how to manipulate and solve linear equations is fundamental in finding the values of x that satisfy the conditions set by the absolute value equations.