# What is the solution set for 4x^2 - 5x < 6?

Aug 7, 2015

Solve $4 {x}^{2} - 5 x < 6$

Ans: $\left(- \frac{3}{4} , 2\right)$

#### Explanation:

Bring the inequality to standard form:
$f \left(x\right) = 4 {x}^{2} - 5 x - 6 < 0$
First, solve $f \left(x\right) = 4 {x}^{2} - 5 x - 6 = 0$ (1) to get the 2 real roots.
I use the new Transforming Method. (Google, Yahoo)
Transformed equation $f ' \left(x\right) = {x}^{2} - 5 x + 24$ (2). Roots have opposite signs.
Factor pairs of 24 -> ...(-2, 12)(-3, 8). This sum is 5 = -b. Then, the 2 real roots of (2) are: -3 and 8.
Back to original equation (1), the 2 real roots are: $- \frac{3}{4}$ and $\frac{8}{4} = 2.$
Find the solution set of the inequality. Since a > 0, the parabola opens upward. Between the 2 real roots $\left(- \frac{3}{4}\right)$ and (2), a part of the parabola is below the x-axis, meaning f(x) < 0.
Answer by open interval:$\left(- \frac{3}{4} , 2\right)$