Solve x-1/2x+5+2x+1/x-1=5/2

$$\frac { x - 1 } { 2 x + 5 } + \frac { 2 x + 1 } { x - 1 } = \frac { 5 } { 2 }$$

Solve for x

$$x = -\frac{37}{5} = -7\frac{2}{5} = -7.4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -\frac{5}{2},1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(2x+5\right), the least common multiple of 2x+5,x-1,2.

$$\left(2x-2\right)\left(x-1\right)+\left(4x+10\right)\left(2x+1\right)=5\left(x-1\right)\left(2x+5\right)$$

Use the distributive property to multiply 2x-2 by x-1 and combine like terms.

$$2x^{2}-4x+2+\left(4x+10\right)\left(2x+1\right)=5\left(x-1\right)\left(2x+5\right)$$

Use the distributive property to multiply 4x+10 by 2x+1 and combine like terms.

$$2x^{2}-4x+2+8x^{2}+24x+10=5\left(x-1\right)\left(2x+5\right)$$

Combine 2x^{2} and 8x^{2} to get 10x^{2}.

$$10x^{2}-4x+2+24x+10=5\left(x-1\right)\left(2x+5\right)$$

Combine -4x and 24x to get 20x.

$$10x^{2}+20x+2+10=5\left(x-1\right)\left(2x+5\right)$$

Add 2 and 10 to get 12.

$$10x^{2}+20x+12=5\left(x-1\right)\left(2x+5\right)$$

Use the distributive property to multiply 5 by x-1.

$$10x^{2}+20x+12=\left(5x-5\right)\left(2x+5\right)$$

Use the distributive property to multiply 5x-5 by 2x+5 and combine like terms.

$$10x^{2}+20x+12=10x^{2}+15x-25$$

Subtract 10x^{2} from both sides.

$$10x^{2}+20x+12-10x^{2}=15x-25$$

Combine 10x^{2} and -10x^{2} to get 0.

$$20x+12=15x-25$$

Subtract 15x from both sides.

$$20x+12-15x=-25$$

Combine 20x and -15x to get 5x.

$$5x+12=-25$$

Subtract 12 from both sides.

$$5x=-25-12$$

Subtract 12 from -25 to get -37.

$$5x=-37$$

Divide both sides by 5.

$$x=\frac{-37}{5}$$

Fraction \frac{-37}{5} can be rewritten as -\frac{37}{5} by extracting the negative sign.

$$x=-\frac{37}{5}$$