Solve a2/2x+11-1/x-4=1/2

$$a \frac { 2 } { 2 x + 11 } - \frac { 1 } { x - 4 } = \frac { 1 } { 2 }$$

Solve for a

$$a=-\frac{\left(2-x\right)\left(2x+11\right)}{4\left(x-4\right)}$$

$$x\neq -\frac{11}{2}\text{ and }x\neq 4$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(x-4\right)\left(2x+11\right), the least common multiple of 2x+11,x-4,2.

$$a\left(2x-8\right)\times 2-\left(4x+22\right)=\left(x-4\right)\left(2x+11\right)$$

Use the distributive property to multiply a by 2x-8.

$$\left(2ax-8a\right)\times 2-\left(4x+22\right)=\left(x-4\right)\left(2x+11\right)$$

Use the distributive property to multiply 2ax-8a by 2.

$$4ax-16a-\left(4x+22\right)=\left(x-4\right)\left(2x+11\right)$$

To find the opposite of 4x+22, find the opposite of each term.

$$4ax-16a-4x-22=\left(x-4\right)\left(2x+11\right)$$

Use the distributive property to multiply x-4 by 2x+11 and combine like terms.

$$4ax-16a-4x-22=2x^{2}+3x-44$$

Add 4x to both sides.

$$4ax-16a-22=2x^{2}+3x-44+4x$$

Combine 3x and 4x to get 7x.

$$4ax-16a-22=2x^{2}+7x-44$$

Add 22 to both sides.

$$4ax-16a=2x^{2}+7x-44+22$$

Add -44 and 22 to get -22.

$$4ax-16a=2x^{2}+7x-22$$

Combine all terms containing a.

$$\left(4x-16\right)a=2x^{2}+7x-22$$

Divide both sides by -16+4x.

$$\frac{\left(4x-16\right)a}{4x-16}=\frac{\left(x-2\right)\left(2x+11\right)}{4x-16}$$

Dividing by -16+4x undoes the multiplication by -16+4x.

$$a=\frac{\left(x-2\right)\left(2x+11\right)}{4x-16}$$

Divide \left(-2+x\right)\left(11+2x\right) by -16+4x.

$$a=\frac{\left(x-2\right)\left(2x+11\right)}{4\left(x-4\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt{16a^{2}-184a+225}}{4}+a-\frac{7}{4}\text{, }&\text{unconditionally}\\x=-\frac{\sqrt{16a^{2}-184a+225}}{4}+a-\frac{7}{4}\text{, }&a\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{16a^{2}-184a+225}}{4}+a-\frac{7}{4}\text{, }&a\geq \sqrt{19}+\frac{23}{4}\text{ or }a\leq \frac{23}{4}-\sqrt{19}\\x=-\frac{\sqrt{16a^{2}-184a+225}}{4}+a-\frac{7}{4}\text{, }&a\geq \sqrt{19}+\frac{23}{4}\text{ or }\left(a\neq 0\text{ and }a\leq \frac{23}{4}-\sqrt{19}\right)\end{matrix}\right.$$