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Chapter 1: Problem 22

Solve the equations. \(6=7+|9 z-3|\)

Short Answer

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Step by step solution

01

Isolate the absolute value

To isolate the absolute value expression, subtract 7 from both sides of the equation: 6 - 7 = |9z - 3| This simplifies to: -1 = |9z - 3|.
02

Understanding absolute value

Recall that the absolute value of a number is always non-negative. Therefore, |9z - 3| must be equal to -1. However, absolute values cannot be negative.
03

Conclude no solution

Since there are no real numbers that satisfy the equation -1 = |9z - 3|, this equation has no solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolation of Absolute Value
Now, we have the absolute value \( |9z - 3| \) isolated, which allows us to analyze it further. Successfully isolating the absolute value term is critical for the next steps of the solution.
Non-Negative Property of Absolute Value
We notice an inconsistency because the left-hand side is negative while the right-hand side, being an absolute value, should be non-negative. Therefore, \(|9z - 3| = -1\) does not hold true under any circumstances.
No Solution Cases in Algebra
In algebra, there are instances where an equation has no valid solutions. Understanding these cases is crucial for correctly solving problems. For absolute value equations, if you end up with an equation where the absolute value is equated to a negative number, no solution exists.
Consider \(-1 = |9z - 3| \). Because absolute values represent distances and cannot be negative, \( |9z - 3| eq -1 \).
Simply put: \( |x| = -y \text{ has no solution for any real number } y > 0. \)
Recognizing these scenarios will help you determine when certain equations cannot be solved.

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