I am working on a problem that requires me to show whether a solution to the system of equations $x^T A x > 0$ and $b^T x < 0$ exists or not, for some $x$. It important to note that $x \in \mathbb{R}^n$, $b \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ and that $b$ and $A$ are known and the question is whether an $x$ exists (so that means that $x$ is unknown) so that both of the conditions ($x^T A x > 0$ and $b^T x < 0$) are satisfied.
Now, I know that if $A$ is positive definite then all non-trivial $x$ satisfy the equation $x^T A x > 0$ or if $A$ is negative definite that no $x$ satisfies the equation $x^T A x > 0$, but what happens in the case where $A$ is neither positive nor negative definite, in addition to that I also have to worry about whether there exists an $x$ that satisfies $x^T A x > 0$ and $b^T x < 0$ simultaneously.
I really have no idea how to approach this problem and any help would be appreciated.
Thanks in advance.
If $b=0$ or if $x^T A x \le 0$ for all $x$, there is obviously no solution. Let us now suppose that $b \neq 0$ and $x_0^T A x_0 > 0$ for some $x_0$. Let $x = x_0+ t b$. When $t$ is close enough to $0$ and different from $0$, one has $x^T A x > 0$ and $b^T x = b^T x_0 + t\ b^T b \neq 0$. If $b^T x < 0$, then $x$ is a solution, otherwise $-x$ is a solution.