For what values of $b$ does the equation $x^2+bx+1=0$ have no real solutions?

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For no real solutions, the discriminant (expression under the square root of the quadratic formula) must be negative.

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ $$b^2-4ac \lt 0$$ $$b^2-4(1)(1) \lt 0$$ $$b^2-4 \lt 0$$ $$b^2 \lt 4$$

The appropriate way to solve this inequality would be to graph $b^2$ and check when it is below the line $y=4$. This occurs from $-2$ to $2$, noninclusive of the endpoints.