(3x^2-2x)/(3x+2)<=-2 — step-by-step solution

⚠Ⓘ 3x is read as the product 3*x: for a power, write 3^x‖2x is read as the product 2*x: for a power, write 2^x

1) Let's solve in

ℝ

\frac{3 x^{2} - 2 x}{3 x + 2} <= -2

\frac{x \cdot (3 x - 2)}{3 x + 2} <= -2

2) Let's add fractions:

\frac{x \cdot (3 x - 2)}{3 x + 2} + 2 <= 0

\frac{x \cdot (3 x - 2) + 2 \cdot (3 x + 2)}{3 x + 2} <= 0

3) Let's solve in

ℝ

x \cdot (3 x - 2) + 2 \cdot (3 x + 2) <= 0

ℹ

x \cdot (3 x - 2)

x^{2} \cdot 3 + x \cdot (-2)

‖

2 \cdot (3 x + 2)

6 x + 4

(x^{2} \cdot 3 + x \cdot (-2)) + (6 x + 4) <= 0

x^{2} \cdot 3

- x \cdot 2

+ 6 x

+ 4 <= 0

3 x^{2} + 4 x + 4 <= 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = 4^{2} - 4 \cdot 3 \cdot 4

ℹ

4^{2}

16

‖

3 \cdot 4

12

Δ = 16 - 12 \cdot 4

Δ = 16 - 48

Δ = -32

Since a =

3

> 0, the solution is:

\emptyset

Could not solve in

ℝ

for:

x

4) Let's solve in

ℝ

3 x + 2 < 0

3 x < -2

Solution found:

x < - \frac{2}{3}

Let's build the table of the different terms to determine the global solution:

▶Solution found:

x \in (-∞, - \frac{2}{3})

5) The numeric value of:

x \in (-∞, - \frac{2}{3})

equals:

▷

x \in (-∞, \approx -0.6667)