Solve 4+x/16+x+b+x/4+x=5/2

$$\frac { 4 + x } { 16 + x } + \frac { b + x } { 4 + x } = \frac { 5 } { 2 }$$

Solve for b

$$b=\frac{x^{2}+52x+288}{2\left(x+16\right)}$$

$$x\neq -16\text{ and }x\neq -4$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(x+4\right)\left(x+16\right), the least common multiple of 16+x,4+x,2.

$$\left(2x+8\right)\left(4+x\right)+\left(2x+32\right)\left(b+x\right)=5\left(x+4\right)\left(x+16\right)$$

Use the distributive property to multiply 2x+8 by 4+x and combine like terms.

$$16x+2x^{2}+32+\left(2x+32\right)\left(b+x\right)=5\left(x+4\right)\left(x+16\right)$$

Use the distributive property to multiply 2x+32 by b+x.

$$16x+2x^{2}+32+2xb+2x^{2}+32b+32x=5\left(x+4\right)\left(x+16\right)$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$16x+4x^{2}+32+2xb+32b+32x=5\left(x+4\right)\left(x+16\right)$$

Combine 16x and 32x to get 48x.

$$48x+4x^{2}+32+2xb+32b=5\left(x+4\right)\left(x+16\right)$$

Use the distributive property to multiply 5 by x+4.

$$48x+4x^{2}+32+2xb+32b=\left(5x+20\right)\left(x+16\right)$$

Use the distributive property to multiply 5x+20 by x+16 and combine like terms.

$$48x+4x^{2}+32+2xb+32b=5x^{2}+100x+320$$

Subtract 48x from both sides.

$$4x^{2}+32+2xb+32b=5x^{2}+100x+320-48x$$

Combine 100x and -48x to get 52x.

$$4x^{2}+32+2xb+32b=5x^{2}+52x+320$$

Subtract 4x^{2} from both sides.

$$32+2xb+32b=5x^{2}+52x+320-4x^{2}$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$32+2xb+32b=x^{2}+52x+320$$

Subtract 32 from both sides.

$$2xb+32b=x^{2}+52x+320-32$$

Subtract 32 from 320 to get 288.

$$2xb+32b=x^{2}+52x+288$$

Combine all terms containing b.

$$\left(2x+32\right)b=x^{2}+52x+288$$

Divide both sides by 32+2x.

$$\frac{\left(2x+32\right)b}{2x+32}=\frac{x^{2}+52x+288}{2x+32}$$

Dividing by 32+2x undoes the multiplication by 32+2x.

$$b=\frac{x^{2}+52x+288}{2x+32}$$

Divide x^{2}+52x+288 by 32+2x.

$$b=\frac{x^{2}+52x+288}{2\left(x+16\right)}$$

Solve for x

$$\left\{\begin{matrix}\\x=-\sqrt{b^{2}-20b+388}+b-26\text{, }&\text{unconditionally}\\x=\sqrt{b^{2}-20b+388}+b-26\text{, }&b\neq 4\end{matrix}\right.$$