Solve 6-7x>9x^2

$$6 - 7 x > 9 x ^ { 2 }$$

Solve for x

$$x\in \left(\frac{-\sqrt{265}-7}{18},\frac{\sqrt{265}-7}{18}\right)$$

Steps Using the Quadratic Formula

Subtract 9x^{2} from both sides.

$$6-7x-9x^{2}>0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 6-7x-9x^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-6+7x+9x^{2}<0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-6+7x+9x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 9 for a, 7 for b, and -6 for c in the quadratic formula.

$$x=\frac{-7±\sqrt{7^{2}-4\times 9\left(-6\right)}}{2\times 9}$$

Do the calculations.

$$x=\frac{-7±\sqrt{265}}{18}$$

Solve the equation x=\frac{-7±\sqrt{265}}{18} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{265}-7}{18} x=\frac{-\sqrt{265}-7}{18}$$

Rewrite the inequality by using the obtained solutions.

$$9\left(x-\frac{\sqrt{265}-7}{18}\right)\left(x-\frac{-\sqrt{265}-7}{18}\right)<0$$

For the product to be negative, x-\frac{\sqrt{265}-7}{18} and x-\frac{-\sqrt{265}-7}{18} have to be of the opposite signs. Consider the case when x-\frac{\sqrt{265}-7}{18} is positive and x-\frac{-\sqrt{265}-7}{18} is negative.

$$x-\frac{\sqrt{265}-7}{18}>0 x-\frac{-\sqrt{265}-7}{18}<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-\frac{-\sqrt{265}-7}{18} is positive and x-\frac{\sqrt{265}-7}{18} is negative.

$$x-\frac{-\sqrt{265}-7}{18}>0 x-\frac{\sqrt{265}-7}{18}<0$$

The solution satisfying both inequalities is x\in \left(\frac{-\sqrt{265}-7}{18},\frac{\sqrt{265}-7}{18}\right).

$$x\in \left(\frac{-\sqrt{265}-7}{18},\frac{\sqrt{265}-7}{18}\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(\frac{-\sqrt{265}-7}{18},\frac{\sqrt{265}-7}{18}\right)$$