solve and check (2x+3)/x + 4/(x-1) =0

Question

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

Answer 1

The equation is (2x + 3)/x + 4/(x - 1) = 0.

Multiply x(x - 1) to each term.

[(2x + 3)/x](x(x - 1)) + 4/(x - 1)(x(x - 1)) = 0(x(x - 1))

Cancel common terms.

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

Apply distributive properety : a (b + c) = ab + ac.

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

Distribute the terms using distributive property.

2x^2 - 2x + 3x - 3 + 4x = 0

2x^2 + 5x - 3 = 0

Factor by grouping method.

2x^2 + 6x - x - 3 = 0

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

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

1st solution

(x + 3) = 0

Subtract 3 from each side.

x = - 3.

2 nd solution

(2x - 1) = 0

Add 1 to each side.

2x = 1

Divide each side by 2.

x = 1/2.

Solution of the equation is x = - 3 and x = 1/2.

Check :

Substitute the value x = - 3 in the original equation.

(2(- 3) + 3)/- 3 + 4/(- 3 - 1) = 0.

1 - 1 = 0

0 = 0.

The above statement is true.so x = - 3 is a solution of the equation.

Substitute the value x = 1/2 in the original equation.

(2(1/2) + 3)/(1/2) + 4/((1/2) - 1) = 0.

8 - 8 = 0

0 = 0.

The above statement is true.so x = 1/2  is a solution of the equation.