Question

Polynomial Inequalities:

9) 2x^2 > 5x + 3

show number line and answer in interval notation

Answer 1

Let us solve the equality $2x^2 > 5x + 3,$ which is equivalent to $2x^2 - 5x - 3>0,$ and hence to $(x-3)(2x+1)>0.$ The equation $(x-3)(2x+1)=0$ has two roots $x_1=3,\ x_2=-\frac{1}2=-0.5.$

Let us use number line test. For this draw the following intervals:

Since $2(-1)^2-5(-1)-3=4>0,\ 2\cdot0^2-5\cdot 0-3=-3<0,$ and $2\cdot 4^2-5\cdot 4-3=9>0,$ we conclude that the solutions of the inequality belongs to the real interval $(-\infty, -0.5)\cup(3,+\infty).$