Finding for n in an inequality

Question

Inequality :

$$\alpha_1 + ( n - 1)a - \bigl(\lfloor n \div m\rfloor \times ( a- b)\bigr) \geq x $$

The following is as far as I get:

$\alpha_1 + ( n - 1)a - \bigl(\lfloor n \div m \rfloor \times ( a- b)\bigr) \geq x $

$(n-1)a - \bigl(\lfloor n\div m\rfloor \times(a-b)\bigr) \ge x - \alpha_1$

$n-1 - \bigl(\lfloor n\div m\rfloor \times(a-b)\bigr) \ge \frac {x-\alpha_1}{a}$

$n - \bigl(\lfloor n\div m\rfloor \times(a-b)\bigr) \ge \frac {x-\alpha_1}{a}$ + 1

This question is also related to this: Formula to find n where nth term >= x

Answer 1

Hint: You can rearrange your inequality to form $\lfloor x\rfloor \le c$ which translates to a condition $c < x \le c+1$ and solve this set of inequalities