3. Fractions: Clear the Denominators

Fractions are not a different kind of algebra. They only make the arithmetic heavier. To simplify, multiply every term by the least common denominator (LCD).

Example 1:
\[ \frac{x}{2}+\frac{x}{3}=10. \]
The LCD of 2 and 3 is 6. Multiply every term by 6:
\[ 6\cdot\frac{x}{2}+6\cdot\frac{x}{3}=6\cdot10. \]
Simplify:
\[ 3x+2x=60. \]
So:
\[ 5x=60,\qquad x=12. \]

Example 2:
\[ \frac{x+1}{3}+\frac{x-2}{2}=7. \]
LCD is 6:
\[ 2(x+1)+3(x-2)=42. \]
Distribute:
\[ 2x+2+3x-6=42. \]
Combine:
\[ 5x-4=42. \]
Solve:
\[ 5x=46,\qquad x=\frac{46}{5}. \]

Example 3:
\[ \frac{2x-1}{5}=\frac{x+4}{3}. \]
Cross-multiply:
\[ 3(2x-1)=5(x+4). \]
Distribute:
\[ 6x-3=5x+20. \]
So:
\[ x=23. \]

Common mistake: multiplying only the fraction terms and forgetting constants. Every term on both sides must be multiplied by the LCD.