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

Verified step by step guidance

1. Understand that the equation involves an absolute value: \(|x+4| = 7\). The absolute value of an expression equals 7 means the expression inside can be either 7 or -7.
2. Set up two separate equations based on the definition of absolute value: \(x + 4 = 7\) and \(x + 4 = -7\).
3. Solve the first equation \(x + 4 = 7\) by isolating \(x\): subtract 4 from both sides to get \(x = 7 - 4\).
4. Solve the second equation \(x + 4 = -7\) by isolating \(x\): subtract 4 from both sides to get \(x = -7 - 4\).
5. Write the solution set as the two values of \(x\) found from the two equations.

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 any expression |A| = B, where B ≥ 0, the solutions are A = B or A = -B. Understanding this helps in setting up equations from absolute value expressions.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side to find its value. After removing the absolute value by considering both positive and negative cases, you solve each resulting linear equation separately to find all possible solutions.

Checking Solutions in Absolute Value Equations

Not all solutions derived algebraically may satisfy the original absolute value equation, especially in inequalities. It is important to substitute solutions back into the original equation to verify their validity and ensure no extraneous solutions are included.