Math worksheet for solving rational expressions with ten algebraic equations.

Problem: Solve each rational equation.

We will solve each equation step by step. Let's go through them one by one.

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#### 1) \(\frac{1}{24x^2} = \frac{1}{12x^2} - \frac{1}{x}\)

1. Find a common denominator for all terms. The denominators are \(24x^2\), \(12x^2\), and \(x\). The least common denominator (LCD) is \(24x^2\).
2. Rewrite the equation with the common denominator:
\[
\frac{1}{24x^2} = \frac{2}{24x^2} - \frac{24x}{24x^2}
\]
3. Combine the terms on the right-hand side:
\[
\frac{1}{24x^2} = \frac{2 - 24x}{24x^2}
\]
4. Since the denominators are the same, equate the numerators:
\[
1 = 2 - 24x
\]
5. Solve for \(x\):
\[
1 - 2 = -24x \implies -1 = -24x \implies x = \frac{1}{24}
\]

Solution: \(x = \frac{1}{24}\)

---

#### 2) \(\frac{1}{n} = \frac{11}{6n} - 8\)

1. The denominators are \(n\) and \(6n\). The LCD is \(6n\).
2. Rewrite the equation with the common denominator:
\[
\frac{6}{6n} = \frac{11}{6n} - \frac{48n}{6n}
\]
3. Combine the terms on the right-hand side:
\[
\frac{6}{6n} = \frac{11 - 48n}{6n}
\]
4. Equate the numerators:
\[
6 = 11 - 48n
\]
5. Solve for \(n\):
\[
6 - 11 = -48n \implies -5 = -48n \implies n = \frac{5}{48}
\]

Solution: \(n = \frac{5}{48}\)

---

#### 3) \(\frac{y - 6}{10y^2} + \frac{12}{5y^2} = \frac{y + 7}{5y^2}\)

1. The denominators are \(10y^2\) and \(5y^2\). The LCD is \(10y^2\).
2. Rewrite the equation with the common denominator:
\[
\frac{y - 6}{10y^2} + \frac{24}{10y^2} = \frac{2(y + 7)}{10y^2}
\]
3. Combine the terms on the left-hand side:
\[
\frac{y - 6 + 24}{10y^2} = \frac{2(y + 7)}{10y^2}
\]
4. Simplify the numerator on the left-hand side:
\[
\frac{y + 18}{10y^2} = \frac{2y + 14}{10y^2}
\]
5. Equate the numerators:
\[
y + 18 = 2y + 14
\]
6. Solve for \(y\):
\[
18 - 14 = 2y - y \implies 4 = y
\]

Solution: \(y = 4\)

---

#### 4) \(\frac{1}{p} + \frac{10p + 7}{p^2 - 3p} = \frac{2p + 4}{p^2 - 3p}\)

1. Factor the denominators where possible:
\[
p^2 - 3p = p(p - 3)
\]
So the denominators are \(p\) and \(p(p - 3)\). The LCD is \(p(p - 3)\).
2. Rewrite the equation with the common denominator:
\[
\frac{p - 3}{p(p - 3)} + \frac{10p + 7}{p(p - 3)} = \frac{2p + 4}{p(p - 3)}
\]
3. Combine the terms on the left-hand side:
\[
\frac{p - 3 + 10p + 7}{p(p - 3)} = \frac{2p + 4}{p(p - 3)}
\]
4. Simplify the numerator on the left-hand side:
\[
\frac{11p + 4}{p(p - 3)} = \frac{2p + 4}{p(p - 3)}
\]
5. Equate the numerators:
\[
11p + 4 = 2p + 4
\]
6. Solve for \(p\):
\[
11p - 2p = 4 - 4 \implies 9p = 0 \implies p = 0
\]
However, \(p = 0\) makes the original denominators undefined. Thus, there is no solution.

Solution: No solution

---

#### 5) \(\frac{1}{d^2} = \frac{1}{25}\)

1. Cross-multiply to solve for \(d^2\):
\[
25 = d^2
\]
2. Take the square root of both sides:
\[
d = \pm 5
\]

Solution: \(d = \pm 5\)

---

#### 6) \(\frac{6}{x - 3} - \frac{1}{x^2 + 6x - 27} = \frac{1}{x - 3}\)

1. Factor the quadratic denominator:
\[
x^2 + 6x - 27 = (x + 9)(x - 3)
\]
So the denominators are \(x - 3\) and \((x + 9)(x - 3)\). The LCD is \((x + 9)(x - 3)\).
2. Rewrite the equation with the common denominator:
\[
\frac{6(x + 9)}{(x + 9)(x - 3)} - \frac{1}{(x + 9)(x - 3)} = \frac{x + 9}{(x + 9)(x - 3)}
\]
3. Combine the terms on the left-hand side:
\[
\frac{6(x + 9) - 1}{(x + 9)(x - 3)} = \frac{x + 9}{(x + 9)(x - 3)}
\]
4. Simplify the numerator on the left-hand side:
\[
\frac{6x + 54 - 1}{(x + 9)(x - 3)} = \frac{x + 9}{(x + 9)(x - 3)}
\]
\[
\frac{6x + 53}{(x + 9)(x - 3)} = \frac{x + 9}{(x + 9)(x - 3)}
\]
5. Equate the numerators:
\[
6x + 53 = x + 9
\]
6. Solve for \(x\):
\[
6x - x = 9 - 53 \implies 5x = -44 \implies x = -\frac{44}{5}
\]

Solution: \(x = -\frac{44}{5}\)

---

#### 7) \(\frac{1}{x - 5} + \frac{1}{x^2 - 11x + 30} = \frac{7}{x - 5}\)

1. Factor the quadratic denominator:
\[
x^2 - 11x + 30 = (x - 5)(x - 6)
\]
So the denominators are \(x - 5\) and \((x - 5)(x - 6)\). The LCD is \((x - 5)(x - 6)\).
2. Rewrite the equation with the common denominator:
\[
\frac{x - 6}{(x - 5)(x - 6)} + \frac{1}{(x - 5)(x - 6)} = \frac{7(x - 6)}{(x - 5)(x - 6)}
\]
3. Combine the terms on the left-hand side:
\[
\frac{x - 6 + 1}{(x - 5)(x - 6)} = \frac{7(x - 6)}{(x - 5)(x - 6)}
\]
4. Simplify the numerator on the left-hand side:
\[
\frac{x - 5}{(x - 5)(x - 6)} = \frac{7(x - 6)}{(x - 5)(x - 6)}
\]
5. Equate the numerators:
\[
x - 5 = 7(x - 6)
\]
6. Solve for \(x\):
\[
x - 5 = 7x - 42 \implies -5 + 42 = 7x - x \implies 37 = 6x \implies x = \frac{37}{6}
\]

Solution: \(x = \frac{37}{6}\)

---

#### 8) \(5 + \frac{x^2 - 32}{7x} = \frac{x + 3}{7x}\)

1. The denominators are \(7x\). Rewrite the equation:
\[
\frac{35x}{7x} + \frac{x^2 - 32}{7x} = \frac{x + 3}{7x}
\]
2. Combine the terms on the left-hand side:
\[
\frac{35x + x^2 - 32}{7x} = \frac{x + 3}{7x}
\]
3. Equate the numerators:
\[
35x + x^2 - 32 = x + 3
\]
4. Rearrange the equation:
\[
x^2 + 35x - x - 32 - 3 = 0 \implies x^2 + 34x - 35 = 0
\]
5. Factor the quadratic equation:
\[
(x + 35)(x - 1) = 0
\]
6. Solve for \(x\):
\[
x = -35 \quad \text{or} \quad x = 1
\]

Solution: \(x = -35\) or \(x = 1\)

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#### 9) \(\frac{1}{16k^2} = \frac{1}{8k^2} - \frac{1}{k}\)

1. The denominators are \(16k^2\), \(8k^2\), and \(k\). The LCD is \(16k^2\).
2. Rewrite the equation with the common denominator:
\[
\frac{1}{16k^2} = \frac{2}{16k^2} - \frac{16k}{16k^2}
\]
3. Combine the terms on the right-hand side:
\[
\frac{1}{16k^2} = \frac{2 - 16k}{16k^2}
\]
4. Equate the numerators:
\[
1 = 2 - 16k
\]
5. Solve for \(k\):
\[
1 - 2 = -16k \implies -1 = -16k \implies k = \frac{1}{16}
\]

Solution: \(k = \frac{1}{16}\)

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#### 10) \(\frac{1}{r} = \frac{5}{9c} - 6\)

1. The denominators are \(r\) and \(9c\). The LCD is \(9cr\).
2. Rewrite the equation with the common denominator:
\[
\frac{9c}{9cr} = \frac{5r}{9cr} - \frac{54cr}{9cr}
\]
3. Combine the terms on the right-hand side:
\[
\frac{9c}{9cr} = \frac{5r - 54cr}{9cr}
\]
4. Equate the numerators:
\[
9c = 5r - 54cr
\]
5. Solve for \(r\):
\[
9c = r(5 - 54c) \implies r = \frac{9c}{5 - 54c}
\]

Solution: \(r = \frac{9c}{5 - 54c}\)

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Final Answers:

\[
\boxed{
\begin{aligned}
1) & \ x = \frac{1}{24} \\
2) & \ n = \frac{5}{48} \\
3) & \ y = 4 \\
4) & \ \text{No solution} \\
5) & \ d = \pm 5 \\
6) & \ x = -\frac{44}{5} \\
7) & \ x = \frac{37}{6} \\
8) & \ x = -35 \text{ or } x = 1 \\
9) & \ k = \frac{1}{16} \\
10) & \ r = \frac{9c}{5 - 54c}
\end{aligned}
}
\]