Solve frac{sqrt{2}x}{sqrt{2}-1}-1=1/sqrt{2+1}+frac{3sqrt{2}+4x-6}{2}

$$\frac { \sqrt { 2 } x } { \sqrt { 2 } - 1 } - 1 = \frac { 1 } { \sqrt { 2 } + 1 } + \frac { 3 \sqrt { 2 } + 4 x - 6 } { 2 }$$

Solve for x

$$x=\frac{5-3\sqrt{2}}{2}\approx 0.378679656$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$\left(2\times 2^{\frac{1}{2}}+2\right)\sqrt{2}x-2=2\times 2^{\frac{1}{2}}-2+3\sqrt{2}+4x-6$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\left(2^{\frac{3}{2}}+2\right)\sqrt{2}x-2=2\times 2^{\frac{1}{2}}-2+3\sqrt{2}+4x-6$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\left(2^{\frac{3}{2}}+2\right)\sqrt{2}x-2=2^{\frac{3}{2}}-2+3\sqrt{2}+4x-6$$

Use the distributive property to multiply 2^{\frac{3}{2}}+2 by \sqrt{2}.

$$\left(2^{\frac{3}{2}}\sqrt{2}+2\sqrt{2}\right)x-2=2^{\frac{3}{2}}-2+3\sqrt{2}+4x-6$$

Use the distributive property to multiply 2^{\frac{3}{2}}\sqrt{2}+2\sqrt{2} by x.

$$2^{\frac{3}{2}}\sqrt{2}x+2\sqrt{2}x-2=2^{\frac{3}{2}}-2+3\sqrt{2}+4x-6$$

Subtract 6 from -2 to get -8.

$$2^{\frac{3}{2}}\sqrt{2}x+2\sqrt{2}x-2=2^{\frac{3}{2}}-8+3\sqrt{2}+4x$$

Subtract 4x from both sides.

$$2^{\frac{3}{2}}\sqrt{2}x+2\sqrt{2}x-2-4x=2^{\frac{3}{2}}-8+3\sqrt{2}$$

Add 2 to both sides.

$$2^{\frac{3}{2}}\sqrt{2}x+2\sqrt{2}x-4x=2^{\frac{3}{2}}-8+3\sqrt{2}+2$$

Add -8 and 2 to get -6.

$$2^{\frac{3}{2}}\sqrt{2}x+2\sqrt{2}x-4x=2^{\frac{3}{2}}-6+3\sqrt{2}$$

Combine all terms containing x.

$$\left(2^{\frac{3}{2}}\sqrt{2}+2\sqrt{2}-4\right)x=2^{\frac{3}{2}}-6+3\sqrt{2}$$

The equation is in standard form.

$$\left(2\sqrt{2}\sqrt{2}+2\sqrt{2}-4\right)x=2\sqrt{2}+3\sqrt{2}-6$$

Divide both sides by 2\times 2^{\frac{1}{2}}\sqrt{2}+2\sqrt{2}-4.

$$\frac{\left(2\sqrt{2}\sqrt{2}+2\sqrt{2}-4\right)x}{2\sqrt{2}\sqrt{2}+2\sqrt{2}-4}=\frac{5\sqrt{2}-6}{2\sqrt{2}\sqrt{2}+2\sqrt{2}-4}$$

Dividing by 2\times 2^{\frac{1}{2}}\sqrt{2}+2\sqrt{2}-4 undoes the multiplication by 2\times 2^{\frac{1}{2}}\sqrt{2}+2\sqrt{2}-4.

$$x=\frac{5\sqrt{2}-6}{2\sqrt{2}\sqrt{2}+2\sqrt{2}-4}$$

Divide 5\sqrt{2}-6 by 2\times 2^{\frac{1}{2}}\sqrt{2}+2\sqrt{2}-4.

$$x=\frac{5-3\sqrt{2}}{2}$$