Solve for f: 5 - 2/f + 3/2f = f - 1/2f + 4.

Steps to Solve

1. Combine like terms on the left side

Start by simplifying the left side of the equation: $$ 5 - \frac{2}{5}f + \frac{3}{2}f $$

Convert $ \frac{2}{5}f $ and $ \frac{3}{2}f $ to a common denominator. The least common multiple of 5 and 2 is 10. Thus: $$ \frac{2}{5}f = \frac{4}{10}f $$ $$ \frac{3}{2}f = \frac{15}{10}f $$

Now substitute: $$ 5 - \frac{4}{10}f + \frac{15}{10}f $$

Combine the like terms: $$ 5 + \frac{15 - 4}{10}f = 5 + \frac{11}{10}f $$

2. Simplify the right side

Now simplify the right side of the equation: $$ f - \frac{1}{2}f + 4 $$

Convert $ \frac{1}{2}f $ to a common denominator of 2: $$ f - \frac{1}{2}f = \frac{2}{2}f - \frac{1}{2}f = \frac{1}{2}f $$

Substituting this back results in: $$ \frac{1}{2}f + 4 $$

3. Set the simplified sides equal to each other

Now equate the simplified left and right sides: $$ 5 + \frac{11}{10}f = \frac{1}{2}f + 4 $$

4. Isolate f on one side

To isolate $f$, first subtract $4$ from both sides: $$ 1 + \frac{11}{10}f = \frac{1}{2}f $$

Next, subtract $\frac{1}{2}f$ from both sides: $$ 1 + \frac{11}{10}f - \frac{5}{10}f = 0 $$

Combine like terms: $$ 1 + \frac{6}{10}f = 0 $$

5. Solve for f

Now, subtract 1 from both sides: $$ \frac{6}{10}f = -1 $$

Multiply both sides by the reciprocal of $\frac{6}{10}$: $$ f = -1 \cdot \frac{10}{6} $$

This simplifies to: $$ f = -\frac{10}{6} = -\frac{5}{3} $$