Solve f(x)=5/3+1/3(3x-1+4/3x-1)

$$f ( x ) = \frac { 5 } { 3 } + \frac { 1 } { 3 } ( 3 x - 1 + \frac { 4 } { 3 x - 1 } )$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x-1 times \frac{3x-1}{3x-1}.

$$\frac{5}{3}+\frac{1}{3}\left(\frac{\left(3x-1\right)\left(3x-1\right)}{3x-1}+\frac{4}{3x-1}\right)$$

Since \frac{\left(3x-1\right)\left(3x-1\right)}{3x-1} and \frac{4}{3x-1} have the same denominator, add them by adding their numerators.

$$\frac{5}{3}+\frac{1}{3}\times \frac{\left(3x-1\right)\left(3x-1\right)+4}{3x-1}$$

Do the multiplications in \left(3x-1\right)\left(3x-1\right)+4.

$$\frac{5}{3}+\frac{1}{3}\times \frac{9x^{2}-3x-3x+1+4}{3x-1}$$

Combine like terms in 9x^{2}-3x-3x+1+4.

$$\frac{5}{3}+\frac{1}{3}\times \frac{9x^{2}-6x+5}{3x-1}$$

Multiply \frac{1}{3} times \frac{9x^{2}-6x+5}{3x-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{3}+\frac{9x^{2}-6x+5}{3\left(3x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 3\left(3x-1\right) is 3\left(3x-1\right). Multiply \frac{5}{3} times \frac{3x-1}{3x-1}.

$$\frac{5\left(3x-1\right)}{3\left(3x-1\right)}+\frac{9x^{2}-6x+5}{3\left(3x-1\right)}$$

Since \frac{5\left(3x-1\right)}{3\left(3x-1\right)} and \frac{9x^{2}-6x+5}{3\left(3x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(3x-1\right)+9x^{2}-6x+5}{3\left(3x-1\right)}$$

Do the multiplications in 5\left(3x-1\right)+9x^{2}-6x+5.

$$\frac{15x-5+9x^{2}-6x+5}{3\left(3x-1\right)}$$

Combine like terms in 15x-5+9x^{2}-6x+5.

$$\frac{9x+9x^{2}}{3\left(3x-1\right)}$$

Factor the expressions that are not already factored in \frac{9x+9x^{2}}{3\left(3x-1\right)}.

$$\frac{9x\left(x+1\right)}{3\left(3x-1\right)}$$

Cancel out 3 in both numerator and denominator.

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

Use the distributive property to multiply 3x by x+1.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x-1 times \frac{3x-1}{3x-1}.

$$\frac{5}{3}+\frac{1}{3}\left(\frac{\left(3x-1\right)\left(3x-1\right)}{3x-1}+\frac{4}{3x-1}\right)$$

Since \frac{\left(3x-1\right)\left(3x-1\right)}{3x-1} and \frac{4}{3x-1} have the same denominator, add them by adding their numerators.

$$\frac{5}{3}+\frac{1}{3}\times \frac{\left(3x-1\right)\left(3x-1\right)+4}{3x-1}$$

Do the multiplications in \left(3x-1\right)\left(3x-1\right)+4.

$$\frac{5}{3}+\frac{1}{3}\times \frac{9x^{2}-3x-3x+1+4}{3x-1}$$

Combine like terms in 9x^{2}-3x-3x+1+4.

$$\frac{5}{3}+\frac{1}{3}\times \frac{9x^{2}-6x+5}{3x-1}$$

Multiply \frac{1}{3} times \frac{9x^{2}-6x+5}{3x-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{3}+\frac{9x^{2}-6x+5}{3\left(3x-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 3\left(3x-1\right) is 3\left(3x-1\right). Multiply \frac{5}{3} times \frac{3x-1}{3x-1}.

$$\frac{5\left(3x-1\right)}{3\left(3x-1\right)}+\frac{9x^{2}-6x+5}{3\left(3x-1\right)}$$

Since \frac{5\left(3x-1\right)}{3\left(3x-1\right)} and \frac{9x^{2}-6x+5}{3\left(3x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(3x-1\right)+9x^{2}-6x+5}{3\left(3x-1\right)}$$

Do the multiplications in 5\left(3x-1\right)+9x^{2}-6x+5.

$$\frac{15x-5+9x^{2}-6x+5}{3\left(3x-1\right)}$$

Combine like terms in 15x-5+9x^{2}-6x+5.

$$\frac{9x+9x^{2}}{3\left(3x-1\right)}$$

Factor the expressions that are not already factored in \frac{9x+9x^{2}}{3\left(3x-1\right)}.

$$\frac{9x\left(x+1\right)}{3\left(3x-1\right)}$$

Cancel out 3 in both numerator and denominator.

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

Use the distributive property to multiply 3x by x+1.

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