Solve 2/3(2x-1)+1/3(2-x)

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

Evaluate

$$x$$

Solution Steps

Use the distributive property to multiply \frac{2}{3} by 2x-1.

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

Express \frac{2}{3}\times 2 as a single fraction.

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

Multiply 2 and 2 to get 4.

$$\frac{4}{3}x+\frac{2}{3}\left(-1\right)+\frac{1}{3}\left(2-x\right)$$

Multiply \frac{2}{3} and -1 to get -\frac{2}{3}.

$$\frac{4}{3}x-\frac{2}{3}+\frac{1}{3}\left(2-x\right)$$

Use the distributive property to multiply \frac{1}{3} by 2-x.

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

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$\frac{4}{3}x-\frac{2}{3}+\frac{2}{3}+\frac{1}{3}\left(-1\right)x$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

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

Add -\frac{2}{3} and \frac{2}{3} to get 0.

$$\frac{4}{3}x-\frac{1}{3}x$$

Combine \frac{4}{3}x and -\frac{1}{3}x to get x.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

Use the distributive property to multiply \frac{2}{3} by 2x-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3}\times 2x+\frac{2}{3}\left(-1\right)+\frac{1}{3}\left(2-x\right))$$

Express \frac{2}{3}\times 2 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\times 2}{3}x+\frac{2}{3}\left(-1\right)+\frac{1}{3}\left(2-x\right))$$

Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x+\frac{2}{3}\left(-1\right)+\frac{1}{3}\left(2-x\right))$$

Multiply \frac{2}{3} and -1 to get -\frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x-\frac{2}{3}+\frac{1}{3}\left(2-x\right))$$

Use the distributive property to multiply \frac{1}{3} by 2-x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x-\frac{2}{3}+\frac{1}{3}\times 2+\frac{1}{3}\left(-1\right)x)$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x-\frac{2}{3}+\frac{2}{3}+\frac{1}{3}\left(-1\right)x)$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x-\frac{2}{3}+\frac{2}{3}-\frac{1}{3}x)$$

Add -\frac{2}{3} and \frac{2}{3} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{3}x-\frac{1}{3}x)$$

Combine \frac{4}{3}x and -\frac{1}{3}x to get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The derivative of ax^{n} is nax^{n-1}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

For any term t except 0, t^{0}=1.

$$1$$