Solve n/2[2(2020)-73(n-1)]<0

$$\frac { n } { 2 } [ 2 ( 2020 ) - 73 ( n - 1 ) ] < 0$$

Solve for n

$$n\in \left(-\infty,0\right)\cup \left(\frac{4113}{73},\infty\right)$$

Solution Steps

Multiply both sides of the equation by 2. Since 2 is positive, the inequality direction remains the same.

$$n\left(2\times 2020-73\left(n-1\right)\right)<0$$

Multiply 2 and 2020 to get 4040.

$$n\left(4040-73\left(n-1\right)\right)<0$$

Use the distributive property to multiply -73 by n-1.

$$n\left(4040-73n+73\right)<0$$

Add 4040 and 73 to get 4113.

$$n\left(4113-73n\right)<0$$

Use the distributive property to multiply n by 4113-73n.

$$4113n-73n^{2}<0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 4113n-73n^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-4113n+73n^{2}>0$$

Factor out n.

$$n\left(73n-4113\right)>0$$

For the product to be positive, n and n-\frac{4113}{73} have to be both negative or both positive. Consider the case when n and n-\frac{4113}{73} are both negative.

$$n<0 n-\frac{4113}{73}<0$$

The solution satisfying both inequalities is n<0.

$$n<0$$

Consider the case when n and n-\frac{4113}{73} are both positive.

$$n-\frac{4113}{73}>0 n>0$$

The solution satisfying both inequalities is n>\frac{4113}{73}.

$$n>\frac{4113}{73}$$

The final solution is the union of the obtained solutions.

$$n<0\text{; }n>\frac{4113}{73}$$