To solve the expression \(2 - \left| \frac{1}{3} x - 9 \right| - 5 = 0\) without absolute value and focusing on the underlying equation, we can rewrite it as:
\[
2 - 5 - \left| \frac{1}{3} x - 9 \right| = 0
\]
This simplifies to:
\[
-3 - \left| \frac{1}{3} x - 9 \right| = 0
\]
From here, we can rearrange it into:
\[
\left| \frac{1}{3} x - 9 \right| = -3
\]
However, since the absolute value cannot equal a negative number, there are no possible solutions.
To explore the absolute value expression in detail, consider the component \(\frac{1}{3} x - 9\):
- Set the inside of the absolute value equal to 3 and -3:
- Case 1: \(\frac{1}{3} x - 9 = 3\)
- Case 2: \(\frac{1}{3} x - 9 = -3\)
Case 1:
\[
\frac{1}{3} x - 9 = 3
\]
Add 9 to both sides:
\[
\frac{1}{3} x = 12
\]
Multiply both sides by 3:
\[
x = 36
\]
Case 2:
\[
\frac{1}{3} x - 9 = -3
\]
Add 9 to both sides:
\[
\frac{1}{3} x = 6
\]
Multiply both sides by 3:
\[
x = 18
\]
Thus, the two cases yield \(x = 36\) and \(x = 18\).
Conclusion:
The original equation gives us no valid solution under the condition of absolute value being negative; therefore, in standard form without contradictions, there is no solution for the condition where the absolute value equals negative values. The exploration gives two distinct solutions from the restructured equations but does not satisfy the original absolute value condition set in the equation.
If you need further clarification or assistance, feel free to ask!
