(x^2-x-2)/(x^2+4x+3)<= 0 — step-by-step solution

1) Let's solve in

ℝ

\frac{x^{2} - x - 2}{x^{2} + 4 x + 3} \leq 0

We study the sign of the numerator

⇥1.1) Let's study the sign of

x^{2} - x - 2

by solving in

ℝ

x^{2} - x - 2 \leq 0

⇥Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -1

and

c = -2

⇥

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-2)

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-1)}^{2}

(-1) \cdot (-1)

1

⇥

Δ = 1 - 4 \cdot (-2)

⇥ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-2)

8

⇥

Δ = 1 + 8

⇥ⓘ Start from

1

and move forward

8

on the number line:

1 + 8

9

⇥

Δ = 9

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{9}}{2 \cdot 1}

⇥

x_{1} = \frac{1 - \sqrt{9}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{9}

3

;

3^{2}

9

⇥

x_{1} = \frac{1 - 3}{2}

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

1 - 3

-2

⇥

x_{1} = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥

x_{1} = -1

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{9}}{2 \cdot 1}

⇥

x_{2} = \frac{1 + \sqrt{9}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{9}

3

;

3^{2}

9

⇥

x_{2} = \frac{1 + 3}{2}

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

1 + 3

4

⇥

x_{2} = \frac{4}{2}

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

2

, their greatest common divisor:

\frac{4}{2}

2

⇥

x_{2} = 2

⇥The quadratic has the opposite sign to a =

1

between its roots, so it is indeed negative between its roots; the roots, where it is zero, are included.

⇥Candidates, pending the domain check:

x \in [-1, 2]

⇥So the expression

x^{2} - x - 2

is negative or zero on

[-1, 2]

and positive otherwise

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

⇥1.2) Let's study the sign of

x^{2} + 4 x + 3

by solving in

ℝ

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

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 4

and

c = 3

⇥

Δ = 4^{2} - 4 \cdot 1 \cdot 3

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{2}

4 \cdot 4

16

⇥

Δ = 16 - 4 \cdot 3

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

⇥

Δ = 16 - 12

⇥ⓘ Start from

16

and move back

12

on the number line:

16 - 12

4

⇥

Δ = 4

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 4 - \sqrt{4}}{2 \cdot 1}

⇥

x_{1} = \frac{- 4 - \sqrt{4}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

⇥

x_{1} = \frac{- 4 - 2}{2}

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

- 4 - 2

-6

⇥

x_{1} = - \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_{1} = -3

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 4 + \sqrt{4}}{2 \cdot 1}

⇥

x_{2} = \frac{- 4 + \sqrt{4}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

⇥

x_{2} = \frac{- 4 + 2}{2}

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

- 4 + 2

-2

⇥

x_{2} = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥

x_{2} = -1

⇥The quadratic has the opposite sign to a =

1

between its roots, so it is indeed negative between its roots; the roots, where it is zero, are excluded.

⇥Candidates, pending the domain check:

x \in (-3, -1)

⇥So the expression

x^{2} + 4 x + 3

is negative on

(-3, -1)

and positive or zero otherwise

2) Let's draw up the sign table:

x	$-∞$		$-3$		$-1$		$2$		$+∞$
$x^{2} - x - 2$		+		+	0	-	0	+
$x^{2} + 4 x + 3$		+	0	-	0	+		+
$\frac{x^{2} - x - 2}{x^{2} + 4 x + 3}$		+	❌	-	❌	-	0	+

Candidates, pending the domain check:

x \in (-3, -1) \cup (-1, 2]

3) Let's find the domain of

\frac{x^{2} - x - 2}{x^{2} + 4 x + 3} \leq 0

For the expression

\frac{x^{2} - x - 2}{x^{2} + 4 x + 3}

to exist, you must check:

x^{2} + 4 x + 3 \neq 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 4

and

c = 3

Δ = 4^{2} - 4 \cdot 1 \cdot 3

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{2}

4 \cdot 4

16

Δ = 16 - 4 \cdot 3

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 3

12

Δ = 16 - 12

ⓘ Start from

16

and move back

12

on the number line:

16 - 12

4

Δ = 4

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 4 - \sqrt{4}}{2 \cdot 1}

x_{1} = \frac{- 4 - \sqrt{4}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

x_{1} = \frac{- 4 - 2}{2}

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

- 4 - 2

-6

x_{1} = - \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_{1} = -3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 4 + \sqrt{4}}{2 \cdot 1}

x_{2} = \frac{- 4 + \sqrt{4}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

x_{2} = \frac{- 4 + 2}{2}

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

- 4 + 2

-2

x_{2} = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

x_{2} = -1

Domain constraints:

x \in ℝ \ {-3, -1}

▶Solution:

x \in (-3, -1) \cup (-1, 2]