What values of 'x' will be the solution to the inequality 15x - 2/x > 1?

May 14, 2016

The answer is x in (-1/3;0) uu (2/5;+oo)

Explanation:

We start with the inequality $15 x - \frac{2}{x} > 1$

First step in solving such inequalities is to determine the domain. We can write that the domain is: $D = \mathbb{R} - \left\{0\right\}$ (all real numbers different from zero).

Next step in solving such (in)equalities is to move all terms to the left side leaving zero on the right side:

$15 x - \frac{2}{x} - 1 > 0$

Now we should write all terms as fractions with comon denominator:

$\frac{15 {x}^{2}}{x} - \frac{2}{x} - \frac{x}{x} > 0$

$\frac{15 {x}^{2} - x - 2}{x} > 0$

Now we have to find zeros of the numerator. To do this we have to calculate the determinant:

$\Delta = 1 - 4 \cdot 15 \cdot \left(- 2\right) = 1 + 120 = 121$

$\sqrt{\Delta} = 11$

${x}_{1} = \frac{1 - 11}{2 \cdot 15} = - \frac{10}{30} = - \frac{1}{3}$

${x}_{2} = \frac{1 + 11}{2 \cdot 15} = \frac{12}{30} = - \frac{2}{5}$

Now we have to sketch the function to find intervals where the values are greater than zero:

graph{x(x+1/3)(x-2/5) [-0.556, 0.556, -0.1, 0.1]}

From this graph we can clearly see the siolution:

x in (-1/3;0) uu (2/5;+oo)