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. Understand that the absolute value equation |x - 2| = 7 can be split into two separate equations because the absolute value of a number is its distance from zero on the number line, which means it can be either positive or negative.
2. Set up the first equation by removing the absolute value and setting the expression inside equal to 7: x - 2 = 7.
3. Solve the first equation for x by adding 2 to both sides: x = 7 + 2.
4. Set up the second equation by removing the absolute value and setting the expression inside equal to -7: x - 2 = -7.
5. Solve the second equation for x by adding 2 to both sides: x = -7 + 2.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases.

Solving Absolute Value Equations

To solve an absolute value equation like |x - 2| = 7, you must consider the two scenarios that arise from the definition of absolute value. This means setting up two separate equations: x - 2 = 7 and x - 2 = -7. Solving these equations will yield the potential solutions for x, which must then be verified in the original equation.

Verification of Solutions

After finding potential solutions from an absolute value equation, it is essential to verify each solution by substituting it back into the original equation. This step ensures that the solutions satisfy the equation, as sometimes extraneous solutions can arise during the solving process. Verification confirms the validity of the solutions found.