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When solving an absolute value equation, the first step is crucial: isolating the absolute value expression on one side of the equation. This means you want to get the absolute value by itself, free of any additional terms. In the exercise given, we start with:\(|3z - 2| + 8 = 1\)To isolate the expression \(|3z - 2|\), we perform inverse operations to remove other terms. Here, we subtract 8 from both sides:\(|3z - 2| = 1 - 8\)
Simplifying this, we obtain:\(|3z - 2| = -7\)Once isolated, you can clearly see the absolute value expression. Isolating allows us to focus on solving the more straightforward absolute value equation. Always remember:

• Use inverse operations like addition or subtraction strategically.
• Keep equations balanced by performing operations equally on both sides.

The concept of the absolute value revolves around its non-negative property. Absolute value denotes the distance of a number from zero on the number line, and distance is always non-negative. Thus, \(|x| \geq 0\)\ for any real number \(x\). Applying this idea to the expression \(|3z - 2| = -7\), the issue becomes apparent:The left-hand side, \(|3z - 2|\), represents absolute value and must be non-negative. Therefore, having it equal to \(-7\), a negative number, creates a contradiction. In simple terms, it's impossible for a distance or absolute value to be less than zero. This non-negative nature of absolute values forms a critical check in solving these equations, and realizing it early saves unnecessary calculations.

Understanding when an equation has no solution is just as important as finding solutions. In the context of absolute value equations, if isolating the expression results in a scenario where the absolute value is set equal to a negative number, you quickly realize there is no solution. In our exercise, we ended with \(|3z - 2| = -7\). The non-negative property of absolute values, discussed earlier, tells us this is impossible. Thus, no real number \(z\) can satisfy the equation. It's essential to recognize:

• When isolated absolute value equals a negative, there's automatically no solution.
• This saves time as you avoid unnecessary steps or calculations.

Identifying situations with no solution hones mathematical intuition and helps tackle similar problems with confidence in the future.