Solve each equation. |7 - 3x| = 3

Verified step by step guidance

1. Start by understanding that the equation \(|7 - 3x| = 3\) involves an absolute value, which means the expression inside the absolute value can be equal to 3 or -3.
2. Set up two separate equations to solve for \(x\): \(7 - 3x = 3\) and \(7 - 3x = -3\).
3. For the first equation \(7 - 3x = 3\), isolate \(x\) by subtracting 7 from both sides, then divide by -3.
4. For the second equation \(7 - 3x = -3\), again isolate \(x\) by subtracting 7 from both sides, then divide by -3.
5. Solve both equations to find the two possible values of \(x\) that satisfy the original absolute value equation.

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 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases based on the definition.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves only linear terms and can be expressed in the form ax + b = c, where a, b, and c are constants. Solving linear equations often involves isolating the variable on one side of the equation. In the context of absolute value equations, each case derived from the absolute value will lead to a linear equation that can be solved for the variable.

Case Analysis

Case analysis is a method used to solve equations that can yield multiple solutions based on different scenarios. For absolute value equations, this involves setting up separate equations for each case derived from the absolute value definition. In the example |7 - 3x| = 3, we create two cases: 7 - 3x = 3 and 7 - 3x = -3, allowing us to find all possible solutions for x.