Solve each equation or inequality. |x+4| = 7

Verified step by step guidance

1. Start by understanding that the equation \(|x+4| = 7\) involves an absolute value, which means the expression inside the absolute value can be either positive or negative.
2. Set up two separate equations to account for both scenarios: \(x + 4 = 7\) and \(x + 4 = -7\).
3. Solve the first equation \(x + 4 = 7\) by subtracting 4 from both sides to isolate \(x\).
4. Solve the second equation \(x + 4 = -7\) by subtracting 4 from both sides to isolate \(x\).
5. Check both solutions by substituting them back into the original equation to ensure they satisfy \(|x+4| = 7\).

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases based on the definition.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In the case of absolute value equations, such as |x + 4| = 7, we set up two separate equations: x + 4 = 7 and x + 4 = -7. This process requires isolating the variable and simplifying to find the solutions.

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like <, >, ≤, or ≥. While the given question focuses on an equation, understanding inequalities is important as they often arise in similar contexts, especially when dealing with absolute values, which can lead to ranges of solutions rather than discrete values.