$\frac{9 z}{9}
eq 9 z$
eq 9 z$
$\frac{9 z}{9}
eq 9 z$
To determine the set of values for \(z\) that satisfy the given inequality \(\frac{9z}{9} eq 9z\), we start by simplifying the expression on the left-hand side. Notice that in the fraction \(\frac{9z}{9}\), the constant factor of 9 in the numerator and the denominator cancels out, simplifying the expression to simply \(z\). Therefore, the inequality \(\frac{9z}{9} eq 9z\) can be written more simply as \(z eq 9z\).
Next, we rearrange the inequality to solve for \(z\). We subtract \(z\) from both sides of the inequality \(z
eq 9z\), which gives:
\[z - z
eq 9z - z\]
\[0
eq 8z\]
To isolate \(z\), we divide both sides by 8, resulting in:
\[z
eq \frac{0}{8}\]
\[z
eq 0\]
This shows that the inequality holds for any value of \(z\) except when \(z\) is equal to zero.
We conclude that the original condition \(\frac{9z}{9} eq 9z\) is satisfied for all complex numbers \(z\) such that \(z eq 0\). If \(z = 0\), the expression becomes \(\frac{9(0)}{9} = 0\) and \(9(0) = 0\), so \(0 eq 0\) is false, confirming that \(z=0\) is the only excluded value. Therefore, the solution set is all \(z \in \mathbb{C} \setminus \{0\}\), or simply \(z eq 0\).
In programming and engineering, verifying that a variable is non-zero is a fundamental step to prevent errors such as division by zero. For instance, if you have a scaling factor \(k\) applied to a signal \(z\), ensuring that \(\frac{k \cdot z}{k} eq k \cdot z\) allows you to identify cases where the scaling operation produces a different result than expected. Understanding these basic algebraic conditions helps in debugging logic where ratios or transformations are involved.