Solve each inequality-write answers in interval notation

Question

(3/(x-1)) greater than or equal to (4/(x+2))

Answer 1

3/(x-1) ≥ 4/(x+2)

3/(x-1) - 4/(x+2) ≥ 0

[3(x+2) - 4(x-1)] / [(x-1)(x+2)] ≥ 0

(10-x) / [(x-1)(x+2)] ≥ 0

Numerator = 0 when x = 10 and Denominator = 0 when x = -2 or 1

Divide the number line into a collection of intervals using -2, 1, and 10 as boundaries.

When x < -2, the numerator and denominator are both positive (try x = -3)

When -2<x<1, the numerator and denominator have opposite signs (try x = 0)

When 1<x<10, the numerator and denominator are both positive (try x = 2)

When x>10, the numerator and denominator have opposite signs (try x = 11)

Solution set: (-∞, -2) ∪ (1,10]

Answer 2

3 / (x - 1) = 4 / (x + 2)

3(x + 2) = 4(x - 1)

3x + 6 = 4x - 4

10 = x

Assume x > 1

3 / (x - 1) > 4 / (x + 2)

3(x + 2) > 4(x - 1)

3x + 6 > 4x - 4

10 > x

Assume x < -2

3 / (x - 1) > 4 / (x + 2)

3(x + 2) < 4(x - 1)

3x + 6 < 4x - 4

10 < x

Assume -2 < x < 1

Can you finish?