The problem is to solve the following equations for $x$: 19. $\frac{x}{2} - \frac{x}{5} = 3$ 20. $\frac{x}{3} = 2 + \frac{x}{4}$ 21. $\frac{5}{x-1} = \frac{10}{x}$ 22. $\frac{12}{2x-3} = 4$ 23. $2 = \frac{18}{x+4}$ 24. $\frac{5}{x+5} = \frac{15}{x+23}$

1. Problem Description

The problem is to solve the following equations for $x$:

1

9. $\frac{x}{2} - \frac{x}{5} = 3$

2

0. $\frac{x}{3} = 2 + \frac{x}{4}$

2

1. $\frac{5}{x-1} = \frac{10}{x}$

2

2. $\frac{12}{2x-3} = 4$

2

3. $2 = \frac{18}{x+4}$

2

4. $\frac{5}{x+5} = \frac{15}{x+23}$

2. Solution Steps

1

9. $\frac{x}{2} - \frac{x}{5} = 3$

Find a common denominator for the fractions, which is

1

0. $\frac{5x}{10} - \frac{2x}{10} = 3$

$\frac{3x}{10} = 3$

Multiply both sides by

1

0. $3x = 30$

Divide both sides by

3. $x = 10$

2

0. $\frac{x}{3} = 2 + \frac{x}{4}$

Multiply both sides by 12 (the least common multiple of 3 and 4).

$12 * \frac{x}{3} = 12 * (2 + \frac{x}{4})$

$4x = 24 + 3x$

Subtract $3x$ from both sides.

$4x - 3x = 24$

$x = 24$

2

1. $\frac{5}{x-1} = \frac{10}{x}$

Cross-multiply.

$5x = 10(x-1)$

$5x = 10x - 10$

Subtract $10x$ from both sides.

$-5x = -10$

Divide both sides by $-5$.

$x = 2$

2

2. $\frac{12}{2x-3} = 4$

Multiply both sides by $(2x-3)$.

$12 = 4(2x-3)$

$12 = 8x - 12$

Add 12 to both sides.

$24 = 8x$

Divide both sides by

8. $x = 3$

2

3. $2 = \frac{18}{x+4}$

Multiply both sides by $(x+4)$.

$2(x+4) = 18$

$2x + 8 = 18$

Subtract 8 from both sides.

$2x = 10$

Divide both sides by

2. $x = 5$

2

4. $\frac{5}{x+5} = \frac{15}{x+23}$

Cross-multiply.

$5(x+23) = 15(x+5)$

$5x + 115 = 15x + 75$

Subtract $5x$ from both sides.

$115 = 10x + 75$

Subtract 75 from both sides.

$40 = 10x$

Divide both sides by

1

0. $x = 4$

3. Final Answer

1

9. $x = 10$

2

0. $x = 24$

2

1. $x = 2$

2

2. $x = 3$

2

3. $x = 5$

2

4. $x = 4$