In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|3x - 2| = 14

Verified step by step guidance

1. Start by isolating the absolute value expression. Divide both sides of the equation by 2 to get: |3x - 2| = 7.
2. Recognize that the absolute value equation |A| = B can be split into two separate equations: A = B and A = -B.
3. Apply this to the equation |3x - 2| = 7, resulting in two equations: 3x - 2 = 7 and 3x - 2 = -7.
4. Solve the first equation, 3x - 2 = 7, by adding 2 to both sides and then dividing by 3 to find the value of x.
5. Solve the second equation, 3x - 2 = -7, by adding 2 to both sides and then dividing by 3 to find the value of 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|, 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 based on the definition.

Solving Absolute Value Equations

To solve an absolute value equation, you must consider both the positive and negative scenarios of the expression inside the absolute value. For the equation |A| = B, where B is non-negative, you set up two separate equations: A = B and A = -B. This approach allows you to find all possible solutions that satisfy the original equation.

Isolating the Absolute Value Expression

Before solving an absolute value equation, it is often necessary to isolate the absolute value expression on one side of the equation. This means rearranging the equation so that it takes the form |expression| = constant. In the given problem, isolating the absolute value helps in applying the definition of absolute value correctly and simplifies the process of finding solutions.