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 with the given equation: \$2|3x - 2| = 14$.
2. Isolate the absolute value expression by dividing both sides of the equation by 2: \(|3x - 2| = \frac{14}{2}\).
3. Simplify the right side: \(|3x - 2| = 7\).
4. Recall that if \(|A| = B\), where \(B > 0\), then \(A = B\) or \(A = -B\). So, set up two equations: \$3x - 2 = 7\( and \)3x - 2 = -7$.
5. Solve each equation separately for \(x\): - For \$3x - 2 = 7\(, add 2 to both sides and then divide by 3. - For \)3x - 2 = -7$, add 2 to both sides and then divide by 3.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A| = B, where B ≥ 0, the solutions satisfy A = B or A = -B. Understanding this is essential to split the equation into two cases for solving.

Solving Linear Equations

After isolating the absolute value expression, solving the resulting linear equations involves applying inverse operations like addition, subtraction, multiplication, and division. This process helps find the values of the variable that satisfy each case derived from the absolute value.

Checking for No Solution

An absolute value equation has no solution if the expression inside the absolute value equals a negative number, since absolute values cannot be negative. It is important to verify the solutions and confirm that the original equation is satisfied.