1. We are given the quadratic equation $$9x^2 + 5x + k = 0$$ and need to find all values of $k$ for which the equation has two real solutions. 2. Recall that a quadratic equation $ax^2 + bx + c = 0$ has two real solutions if and only if its discriminant $$\Delta = b^2 - 4ac$$ is strictly greater than zero. 3. For our equation, $a = 9$, $b = 5$, and $c = k$. Substitute these into the discriminant formula: $$\Delta = 5^2 - 4 \times 9 \times k = 25 - 36k$$ 4. Set the discriminant greater than zero to ensure two real solutions: $$25 - 36k > 0$$ 5. Solve the inequality for $k$: $$25 > 36k$$ $$\frac{25}{36} > k$$ 6. Therefore, the quadratic equation has two real solutions if and only if: $$k < \frac{25}{36}$$ This is the required inequality for $k$. Final answer: $$k < \frac{25}{36}$$