Solve the inequality algebraically, precalc

Question

Solve the inequality and give your answer in interval notation

1+2/(x+1)<=2/x

The explanation is more important than the answer to me. Thanks.

Answer 1

1 + 2/(x+1) - 2/x ≤ 0

[x(x+1) + 2x - 2(x+1)] / [x(x+1)] ≤ 0

(x²+x-2) / [x(x+1)] ≤ 0

[(x+2)(x-1)] / [x{x+1}] ≤ 0

Numerator = 0 when x = -2 and 1. Denominator = 0 when x = 0 and -1.

Set up the following collection of intervals ____-2_____-1____0_____1_______and choose a test point in each interval.

When x < -2, numerator and denominator are both positive. So, the expression is positive.

When -2<x<-1, numerator < 0 and denominator > 0. So, the expression is negative

When -1<x<0, numerator <0 and denominator < 0. So, the expression is positive

When 0<x<1, numerator < 0 and denominator > 0. So, the expression is negative,

When x > 1, the expression is positive

The expression = 0 when x = -2 and 1

The expression is undefined when x = 0 and x = -1

Solution set: [-2, -1)∪(0,1]