Can someone explain how to solve equation number 6? I've tried to solve it multiple times and failed. The answers are: x=1, x=-4.?

1 / (x * (x + 1)) + 1 / ((x + 1) * (x + 2)) + 1 / ((x + 2) * (x + 3)) = 3/4

I'd decompose this into partial fractions, if possible. It might simplify things down the road

A/x + B/(x + 1) = 1/(x * (x + 1))

A * (x + 1) + Bx = 0x + 1

Ax + Bx + A = 0x + 1

A = 1

B = -1

1/x - 1/(x + 1) = 1/(x * (x + 1))

1/((x + 1) * (x + 2)) = A/(x + 1) + B/(x + 2)

A * (x + 2) + B * (x + 1) = 0x + 1

Ax + Bx + 2A + B = 0x + 1

A + B = 0

2A + B = 1

2A + B - A - B = 1 - 0

A = 1

B = -1

1/(x + 1) - 1/(x + 2)

1/((x + 2) * (x + 3)) = A/(x + 2) + B/(x + 3)

A * (x + 3) + B * (x + 2) = 0x + 1

Ax + 3A + Bx + 2B = 0x + 1

A + B = 0

3A + 2B = 1

2A + 2B = 0

3A + 2B - 2A - 2B = 1 - 0

A = 1

B = -1

1/(x + 2) - 1/(x + 3)

Now we have

1/x - 1/(x + 1) + 1/(x + 1) - 1/(x + 2) + 1/(x + 2) - 1/(x + 3) = 3/4

Notice how most of the terms cancel out

1/x - 1/(x + 3) = 3/4

That's much nicer to look at

(1 * (x + 3) - 1 * x) / (x * (x + 3)) = 3 / 4

4 * (x + 3 - x) = 3 * (x^2 + 3x)

4 * 3 = 3 * (x^2 + 3x)

4 = x^2 + 3x

0 = x^2 + 3x - 4

x = (-3 +/- sqrt(9 + 16)) / 2

x = (-3 +/- 5) / 2

x = -8/2 , 2/2

x = -4 , 1

[x/(x + 1)] + [1/(x + 1).(x + 2)] + [1/(x + 2).(x + 3)] = 3/4 → where: x ≠ - 1 and: x ≠ - 2 and: x ≠ - 3

[x.(x + 2).(x + 3) + (x + 3) + (x + 1)] / [(x + 1).(x + 2).(x + 3)] = 3/4

[x.(x + 2).(x + 3) + 2x + 4] / [(x + 1).(x + 2).(x + 3)] = 3/4

[x.(x + 2).(x + 3) + 2.(x + 2)] / [(x + 1).(x + 2).(x + 3)] = 3/4

(x + 2).[x.(x + 3) + 2] / [(x + 1).(x + 2).(x + 3)] = 3/4 → as (x + 2) ≠ 0 → you can simplify by (x + 2)

[x.(x + 3) + 2] / [(x + 1).(x + 3)] = 3/4

[x² + 3x + 2] / [(x + 1).(x + 3)] = 3/4

[x² + (2x + x) + 2] / [(x + 1).(x + 3)] = 3/4

[x² + 2x + x + 2] / [(x + 1).(x + 3)] = 3/4

[(x² + 2x) + (x + 2)] / [(x + 1).(x + 3)] = 3/4

[x.(x + 2) + (x + 2)] / [(x + 1).(x + 3)] = 3/4

[(x + 2).(x + 1)] / [(x + 1).(x + 3)] = 3/4 → as (x + 1) ≠ 0 → you can simplify by (x + 1)

(x + 2) / (x + 3) = 3/4 → you make the cross-multiply

4.(x + 2) = 3.(x + 3)

4x + 8 = 3x + 9

4x - 3x = 9 - 8

x = 1

Instead of multiplying through by the LCD, combine the fractions as we may be able to simplify the result:

[(x + 2)(x + 3) + x(x + 3) + x(x + 1)] / x(x + 1)(x + 2)(x + 3) = 3/4

(x^2 + 5x + 6 + x^2 + 3x + x^2 + x) / x(x + 1)(x + 2)(x + 3) = 3/4

(3x^2 + 9x + 6) / x(x + 1)(x + 2)(x + 3) = 3/4

Factor the numerator:

3(x + 1)(x + 2) / x(x + 1)(x + 2)(x + 3) = 3/4

Simplify:

3 / x(x + 3) = 3/4

1 / x(x + 3) = 1/4

x(x + 3) = 4

Expand the brackets and move everything to one side:

x^2 + 3x - 4 = 0

(x + 4)(x - 1) = 0

x = -4, 1

1 / (x (x + 1)) + 1 / ((x + 1) (x + 2)) + 1 / ((x + 2) (x + 3)) = 3/4

4 / (x (x + 1)) + 4 / ((x + 1) (x + 2)) + 4 / ((x + 2) (x + 3)) = 3

4(x + 2)(x + 3) + 4x(x + 3) + 4x(x + 1) = 3x(x + 1)(x + 2)(x + 3)

4x^2 + 20x + 24 + 4x^2 + 12x + 4x^2 + 4x = (3x^2 + 3x)(x^2 + 5x + 6)

12x^2 + 36x + 24 = 3x^4 + 15x^3 + 18x^2 + 3x^3 + 15x^2 + 18x

3x^4 + 18x^3 + 21x^2 - 18x - 24 = 0

3(x^4 + 6x^3 + 7x^2 - 6x - 8) = 0

3(x - 1) (x + 1) (x + 2) (x + 4) = 0

Solutions:

x = -4

x = -2

x = -1

x = 1

It is correct, the answers are: x=1, x=-4.