In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x - 2| = 7

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|x - 2| = 7\). The absolute value expression \(|A| = B\) means that \(A = B\) or \(A = -B\), provided \(B \geq 0\).
2. Set up two separate equations based on the definition of absolute value: \(x - 2 = 7\) and \(x - 2 = -7\).
3. Solve the first equation \(x - 2 = 7\) by adding 2 to both sides: \(x = 7 + 2\).
4. Solve the second equation \(x - 2 = -7\) by adding 2 to both sides: \(x = -7 + 2\).
5. Write the solution set as the values of \(x\) found from both equations.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always expressed as a non-negative value. For any real number x, |x| equals x if x is non-negative, and -x if x is negative. This concept is crucial for understanding how to solve equations involving absolute values.

Solving Absolute Value Equations

To solve an equation like |x - a| = b, where b is positive, split it into two separate linear equations: x - a = b and x - a = -b. This approach accounts for both possible values inside the absolute value that yield the same distance b from zero.

Checking for No Solution Cases

If the absolute value equation is set equal to a negative number, such as |x - 2| = -7, it has no solution because absolute values cannot be negative. Recognizing this helps avoid unnecessary calculations and correctly identify when no solutions exist.