x/(x-6)>=-1 — step-by-step solution

1) Let's solve in

ℝ

\frac{x}{x - 6} \geq -1

2) Let's sum the fractions:

\frac{x}{x - 6} + 1 \geq 0

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{x}{x - 6} + 1

\frac{x + (x - 6)}{x - 6}

\frac{x + (x - 6)}{x - 6} \geq 0

ⓘ Like terms are grouped by adding their coefficients:

x + x

2 x

\frac{2 x - 6}{x - 6} \geq 0

We study the sign of the numerator

⇥The expression

2 x - 6

is zero at:

⇥

x = \frac{6}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{6}{2}

3

⇥

x = 3

⇥Its coefficient of

x

2

, being positive, it is negative before and positive after

3

We study the sign of the denominator, which must not be zero

⇥The expression

x - 6

is zero at

x = 6

(forbidden value), it is negative before and positive after

3) Let's draw up the sign table:

Candidates, pending the domain check:

x \in (-∞, 3] \cup (6, +∞)

4) Let's evaluate the domain of

\frac{x}{x - 6} \geq -1

For the expression

\frac{x}{x - 6}

to be defined, we must have:

x - 6 \neq 0

ⓘ We add

6

to both sides

Domain constraints:

x \neq 6

▶Solution:

x \in (-∞, 3] \cup (6, +∞)