Solve 2(x+1)-3(x+1)/x-5geq0

$$\frac { 2 ( x + 1 ) - 3 ( x + 1 ) } { x - 5 } \geq 0$$

Solve for x

$$x\in [-1,5)$$

Solution Steps

Combine 2\left(x+1\right) and -3\left(x+1\right) to get -\left(x+1\right).

$$\frac{-\left(x+1\right)}{x-5}\geq 0$$

To find the opposite of x+1, find the opposite of each term.

$$\frac{-x-1}{x-5}\geq 0$$

For the quotient to be ≥0, -x-1 and x-5 have to be both ≤0 or both ≥0, and x-5 cannot be zero. Consider the case when -x-1\leq 0 and x-5 is negative.

$$-x-1\leq 0 x-5<0$$

The solution satisfying both inequalities is x\in \left[-1,5\right).

$$x\in [-1,5)$$

Consider the case when -x-1\geq 0 and x-5 is positive.

$$-x-1\geq 0 x-5>0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in [-1,5)$$