(x-4)/(x+6)<1 — step-by-step solution

1) Let's solve in

ℝ

\frac{x - 4}{x + 6} < 1

2) Let's sum the fractions:

\frac{x - 4}{x + 6} - 1 < 0

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

\frac{x - 4}{x + 6} - 1

\frac{(x - 4) - 1 \cdot (x + 6)}{x + 6}

\frac{(x - 4) - 1 \cdot (x + 6)}{x + 6} < 0

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot (x + 6)

- (x + 6)

\frac{x - 4 - (x + 6)}{x + 6} < 0

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (x + 6)

- x - 6

\frac{x - 4 - x - 6}{x + 6} < 0

ⓘ The numerator and the denominator are computed before the division:

- 4 - 6

-10

■Like terms are grouped by adding their coefficients:

x - x

0

\frac{0 - 10}{x + 6} < 0

ⓘ Adding 0 changes nothing:

0 - 10

-10

- \frac{10}{x + 6} < 0

⇥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 > -6

4) Let's find the domain of

\frac{x - 4}{x + 6} < 1

For the expression

\frac{x - 4}{x + 6}

to exist, you must check:

x + 6 \neq 0

ⓘ We subtract

6

from both sides

Domain constraints:

x \neq -6

▶Solution:

x > -6