In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| = 5

Verified step by step guidance

1. Start by understanding that the absolute value equation |2x - 1| = 5 means that the expression inside the absolute value, 2x - 1, can be either 5 or -5.
2. Set up the first equation by removing the absolute value and setting the expression equal to 5: 2x - 1 = 5.
3. Solve the first equation for x by adding 1 to both sides to get 2x = 6, then divide both sides by 2 to find x.
4. Set up the second equation by removing the absolute value and setting the expression equal to -5: 2x - 1 = -5.
5. Solve the second equation for x by adding 1 to both sides to get 2x = -4, then divide both sides by 2 to find x.

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|. 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 based on the definition.

Solving Absolute Value Equations

To solve an absolute value equation like |2x - 1| = 5, you must consider two scenarios: one where the expression inside the absolute value is equal to the positive value (2x - 1 = 5) and another where it equals the negative value (2x - 1 = -5). This approach allows you to find all possible solutions for the variable.

Checking Solutions

After finding potential solutions from an absolute value equation, it is essential to check each solution by substituting it back into the original equation. This step ensures that the solutions are valid and satisfy the equation, as sometimes extraneous solutions can arise during the solving process.