Solve 32n+n(n-1)/2*(-3)<0

$$32 n + \frac { n ( n - 1 ) } { 2 } \times ( - 3 ) < 0$$

Solve for n

$$n\in \left(-\infty,0\right)\cup \left(\frac{67}{3},\infty\right)$$

Solution Steps

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

$$64n+n\left(n-1\right)\left(-3\right)<0$$

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

$$64n+\left(n^{2}-n\right)\left(-3\right)<0$$

Use the distributive property to multiply n^{2}-n by -3.

$$64n-3n^{2}+3n<0$$

Combine 64n and 3n to get 67n.

$$67n-3n^{2}<0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 67n-3n^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-67n+3n^{2}>0$$

Factor out n.

$$n\left(3n-67\right)>0$$

For the product to be positive, n and n-\frac{67}{3} have to be both negative or both positive. Consider the case when n and n-\frac{67}{3} are both negative.

$$n<0 n-\frac{67}{3}<0$$

The solution satisfying both inequalities is n<0.

$$n<0$$

Consider the case when n and n-\frac{67}{3} are both positive.

$$n-\frac{67}{3}>0 n>0$$

The solution satisfying both inequalities is n>\frac{67}{3}.

$$n>\frac{67}{3}$$

The final solution is the union of the obtained solutions.

$$n<0\text{; }n>\frac{67}{3}$$