Let's solve the problems step by step:

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Problem 15

Given the expression:
$\frac{u - 4}{8v} + \frac{6v - 3x}{8v}$

Combine the fractions since the denominators are the same ($8v$):
$\frac{(u - 4) + (6v - 3x)}{8v}$

Simplify the numerator:
$u - 4 + 6v - 3x$

This gives:
$\frac{u + 6v - 3x - 4}{8v}$

From the given form:
$\frac{a + b}{8v}$

We identify:

• $a = -4$
• $b = -3$

Answer:
$a = -4, \, b = -3$

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Problem 16

Given the expression:
$\frac{r + 6}{3r - 6} + \frac{r + 1}{3r - 6}$

Combine the fractions since the denominators are the same ($3r - 6$):
$\frac{(r + 6) + (r + 1)}{3r - 6}$

Simplify the numerator:
$r + 6 + r + 1 = 2r + 7$

This gives:
$\frac{2r + 7}{3r - 6}$

From the given form:
$\frac{ar + b}{3r - 6}$

We identify:

• $a = 2$
• $b = 7$

Answer:
$a = 2, \, b = 7$

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Problem 17

Given the expression:
$\frac{6}{x - 1} + \frac{5x}{x^2 - 1}$

Factorize the denominator $x^2 - 1$ as $(x - 1)(x + 1)$. Rewrite:
$\frac{6}{x - 1} + \frac{5x}{(x - 1)(x + 1)}$

To combine, find the common denominator $(x - 1)(x + 1)$:

1. Multiply the first fraction by $(x + 1)$:
   $\frac{6(x + 1)}{(x - 1)(x + 1)} = \frac{6x + 6}{(x - 1)(x + 1)}$

2. Add the second fraction:
   $\frac{6x + 6}{(x - 1)(x + 1)} + \frac{5x}{(x - 1)(x + 1)} = \frac{6x + 6 + 5x}{(x - 1)(x + 1)}$

Simplify the numerator:
$6x + 5x + 6 = 11x + 6$

The answer is:
$\frac{11x + 6}{x^2 - 1}$

From the given form:
$\frac{ax^2 + bx + c}{x^2 - 4}$

Comparing terms:

• $a = 0$ (no $x^2$ term)
• $b = 11$
• $c = 6$

Answer:
$a = 0, \, b = 11, \, c = 6$

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Final Answers:

1. Problem 15: $a = -4$, $b = -3$
2. Problem 16: $a = 2$, $b = 7$
3. Problem 17: $a = 0$, $b = 11$, $c = 6$

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Do you want further details or have any questions?

Here are some related practice problems:

1. Simplify: $\frac{2x + 3}{x - 4} + \frac{3x - 2}{x - 4}$
2. Combine: $\frac{5}{x} + \frac{x + 2}{x^2}$
3. Solve: $\frac{4}{x + 2} + \frac{3x}{(x + 2)(x + 3)}$
4. Factorize $x^2 - 4x + 4$ and simplify: $\frac{x^2 - 4x + 4}{x - 2}$
5. Simplify: $\frac{2x + 6}{x + 3} - \frac{x}{x + 3}$

Tip: Always find a common denominator when adding fractions with algebraic terms.