Solve 1/2(2x-2)+3x=10(2/5x-1/10)

$$\frac { 1 } { 2 } ( 2 x - 2 ) + 3 x = 10 ( \frac { 2 } { 5 } x - \frac { 1 } { 10 } )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2} by 2x-2.

$$\frac{1}{2}\times 2x+\frac{1}{2}\left(-2\right)+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Cancel out 2 and 2.

$$x+\frac{1}{2}\left(-2\right)+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Multiply \frac{1}{2} and -2 to get \frac{-2}{2}.

$$x+\frac{-2}{2}+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Divide -2 by 2 to get -1.

$$x-1+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Combine x and 3x to get 4x.

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

Use the distributive property to multiply 10 by \frac{2}{5}x-\frac{1}{10}.

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

Express 10\times \frac{2}{5} as a single fraction.

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

Multiply 10 and 2 to get 20.

$$4x-1=\frac{20}{5}x+10\left(-\frac{1}{10}\right)$$

Divide 20 by 5 to get 4.

$$4x-1=4x+10\left(-\frac{1}{10}\right)$$

Cancel out 10 and 10.

$$4x-1=4x-1$$

Subtract 4x from both sides.

$$4x-1-4x=-1$$

Combine 4x and -4x to get 0.

$$-1=-1$$

Compare -1 and -1.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2} by 2x-2.

$$\frac{1}{2}\times 2x+\frac{1}{2}\left(-2\right)+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Cancel out 2 and 2.

$$x+\frac{1}{2}\left(-2\right)+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Multiply \frac{1}{2} and -2 to get \frac{-2}{2}.

$$x+\frac{-2}{2}+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Divide -2 by 2 to get -1.

$$x-1+3x=10\left(\frac{2}{5}x-\frac{1}{10}\right)$$

Combine x and 3x to get 4x.

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

Use the distributive property to multiply 10 by \frac{2}{5}x-\frac{1}{10}.

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

Express 10\times \frac{2}{5} as a single fraction.

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

Multiply 10 and 2 to get 20.

$$4x-1=\frac{20}{5}x+10\left(-\frac{1}{10}\right)$$

Divide 20 by 5 to get 4.

$$4x-1=4x+10\left(-\frac{1}{10}\right)$$

Cancel out 10 and 10.

$$4x-1=4x-1$$

Subtract 4x from both sides.

$$4x-1-4x=-1$$

Combine 4x and -4x to get 0.

$$-1=-1$$

Compare -1 and -1.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$