Solve the inequality 2x² + x - 1 > 0.

Step-by-step solution

1. 1

   Factor the quadratic. Split the middle term using two numbers with product $2\cdot(-1)=-2$ and sum $+1$, namely $+2$ and $-1$:

   $2x^2+2x-x-1=2x(x+1)-1(x+1)=(2x-1)(x+1)$

   Factoring first is worth the effort: the roots then appear without any square roots.

2. 2

   Read off the roots. $(2x-1)(x+1)=0$ gives $x=\frac12$ and $x=-1$. These split the line into $(-\infty,-1)$, $\left(-1,\frac12\right)$ and $\left(\frac12,\infty\right)$.

3. 3

   Predict the answer from the shape. The leading coefficient $2>0$, so the parabola opens upward: it is positive outside its two roots and negative strictly between them. That already gives the shape of the answer.

4. 4

   Confirm with one test point per interval.

   $x=-2:\ (-5)(-1)=5>0\qquad x=0:\ (-1)(1)=-1<0\qquad x=1:\ (1)(2)=2>0$

   Matching the prediction: the outer intervals work, the middle one does not. Structurally, the product is positive when both factors share a sign.

5. 5

   Handle the endpoints. At $x=-1$ and $x=\frac12$ the expression is exactly $0$, and $0>0$ is false, so the inequality is strict at both ends:

   $x\in(-\infty,-1)\cup\left(\tfrac12,\infty\right)$