## Solving the Equation: 1 4/c - 3 1/2c + 3 = -3 4/c

This equation involves fractions and variables, making it a bit more challenging to solve. Here's a step-by-step guide to find the solution:

### 1. Convert Mixed Numbers to Improper Fractions

**1 4/c:**This becomes (1*c + 4)/c = (c+4)/c**3 1/2c:**This becomes (3*2c + 1)/2 = (6c+1)/2**-3 4/c:**This becomes (-3*c - 4)/c = (-3c-4)/c

### 2. Rewrite the Equation with Improper Fractions

The equation now looks like this: (c+4)/c - (6c+1)/2 + 3 = (-3c-4)/c

### 3. Find a Common Denominator

To combine the terms, we need a common denominator. The least common denominator for c, 2, and c is 2c.

**(c+4)/c:**Multiply numerator and denominator by 2: 2(c+4)/(2c) = (2c+8)/(2c)**(6c+1)/2:**Multiply numerator and denominator by c: c(6c+1)/(2c) = (6c^2+c)/(2c)**3:**Multiply numerator and denominator by 2c: 3(2c)/(2c) = (6c)/(2c)

### 4. Combine Terms

Now the equation looks like this: (2c+8)/(2c) - (6c^2+c)/(2c) + (6c)/(2c) = (-3c-4)/c

Since the denominators are all the same, we can combine the numerators: (2c + 8 - 6c^2 - c + 6c) / (2c) = (-3c - 4)/c

Simplify the numerator: (-6c^2 + 7c + 8) / (2c) = (-3c - 4)/c

### 5. Multiply Both Sides by the Denominator

To eliminate the fractions, multiply both sides of the equation by 2c: -6c^2 + 7c + 8 = -6c - 8

### 6. Move All Terms to One Side

Add 6c and 8 to both sides to set the equation to zero: -6c^2 + 13c + 16 = 0

### 7. Solve the Quadratic Equation

This quadratic equation can be solved using the quadratic formula: c = (-b ± √(b^2 - 4ac)) / (2a)

Where:

- a = -6
- b = 13
- c = 16

Substitute these values into the quadratic formula and solve for c. You will get two possible solutions for c.

**Important Note:** Always check your solutions by plugging them back into the original equation to make sure they are valid.