Solve the inequality 36 - 36k^2 > 0.

Step-by-step solution

1. 1

   Factor out the common positive constant.

   $36\left(1-k^2\right)>0$

   Dividing both sides by $36$ is safe and leaves the direction unchanged because $36>0$:

   $1-k^2>0$

2. 2

   Rearrange into a comparison of squares. Add $k^2$ to both sides:

   $k^2<1$

3. 3

   Translate $k^2<1$ correctly. Taking a square root of an inequality is not a symbol-for-symbol operation; the correct reading is via absolute value:

   $k^2<1\iff|k|<1\iff -1<k<1$

   Writing $k<1$ alone would wrongly admit $k=-5$, whose square is $25$.

4. 4

   Cross-check by factoring instead. $1-k^2=(1-k)(1+k)$, which is positive exactly when both factors share a sign — and that happens only on $-1<k<1$ $\checkmark$.

5. 5

   Test the regions. At $k=0$: $36>0$ $\checkmark$. At $k=2$: $36-144=-108\not>0$ $\checkmark$. At $k=\pm 1$: $36-36=0$, not $>0$, so the endpoints are excluded. This inequality is exactly the condition '$\Delta>0$' in many discriminant problems, where $36-36k^2$ appears as $b^2-4ac$.