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

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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 $B \geq 0$.

Set up two separate equations based on the definition of absolute value: $x - 2 = 7$ and $x - 2 = -7$.

Solve the first equation $x - 2 = 7$ by adding 2 to both sides: $x = 7 + 2$.

Solve the second equation $x - 2 = -7$ by adding 2 to both sides: $x = -7 + 2$.

Write the solution set as the values of $x$ found from both equations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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