(-5x(x+3))/(x-4)>0 — step-by-step solution

1) Let's solve in

ℝ

\frac{-5 x \cdot (x + 3)}{x - 4} > 0

- \frac{5 x \cdot (x + 3)}{x - 4} > 0

ⓘ The minus sign of the fraction goes into the numerator or the denominator, changing the sign of each of its terms:

- \frac{5 x \cdot (x + 3)}{x - 4}

\frac{5 x \cdot (x + 3)}{- x + 4}

\frac{5 x \cdot (x + 3)}{- x + 4} > 0

We study the sign of the numerator

Let's solve in

ℝ

5 x \cdot (x + 3) > 0

We simplify the expression by

5

We study the sign of each factor of the product

⇥The expression

x

is zero at

x = 0

, it is negative before and positive after

⇥The expression

x + 3

is zero at

x = -3

, it is negative before and positive after

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

⇥The expression

- x + 4

is zero at:

⇥

x = 4

⇥Its coefficient of

x

-1

, being negative, it is positive before and negative after

4

(forbidden value)

2) Let's draw up the sign table:

Candidates, pending the domain check:

x \in (-∞, -3) \cup (0, 4)

3) Let's evaluate the domain of

\frac{-5 x \cdot (x + 3)}{x - 4} > 0

For the expression

\frac{-5 x \cdot (x + 3)}{x - 4}

to be defined, we must have:

x - 4 \neq 0

ⓘ We add

4

to both sides

Domain constraints:

x \neq 4

▶Solution:

x \in (-∞, -3) \cup (0, 4)