7.5 solving rational equations
solve each equation. remember to check for
1) $\frac{1}{6k^2} = \frac{1}{3k^2} - \frac{1}{k}$
3) $\frac{1}{6b^2} + \frac{1}{6b} = \frac{1}{b^2}$
5) $\frac{1}{x} = \frac{6}{5x} + 1$
7) $\frac{1}{v} + \frac{3v + 12}{v^2 - 5v} = \frac{7v - 56}{v^2 - 5v}$
9) $\frac{1}{n - 8} - 1 = \frac{7}{n - 8}$

Question

Accepted Answer

### Problem 1: Solve $\boldsymbol{\frac{1}{6k^2} = \frac{1}{3k^2} - \frac{1}{k}}$
# Explanation:
## Step1: Find common denominator ($6k^2$)
Multiply each term by $6k^2$ to eliminate denominators:
$6k^2 \cdot \frac{1}{6k^2} = 6k^2 \cdot \frac{1}{3k^2} - 6k^2 \cdot \frac{1}{k}$
Simplify: $1 = 2 - 6k$

## Step2: Solve for $k$
Subtract 2 from both sides: $1 - 2 = -6k$
$-1 = -6k$
Divide by $-6$: $k = \frac{1}{6}$

## Step3: Check for extraneous solutions
Denominators: $6k^2
eq 0$, $3k^2
eq 0$, $k
eq 0$. $k = \frac{1}{6}$ is valid.
# Answer: $k = \frac{1}{6}$

### Problem 3: Solve $\boldsymbol{\frac{1}{6b^2} + \frac{1}{6b} = \frac{1}{b^2}}$
# Explanation:
## Step1: Common denominator ($6b^2$)
Multiply all terms by $6b^2$:
$6b^2 \cdot \frac{1}{6b^2} + 6b^2 \cdot \frac{1}{6b} = 6b^2 \cdot \frac{1}{b^2}$
Simplify: $1 + b = 6$

## Step2: Solve for $b$
Subtract 1: $b = 5$

## Step3: Check validity
Denominators: $6b^2
eq 0$, $6b
eq 0$, $b^2
eq 0$. $b = 5$ is valid.
# Answer: $b = 5$

### Problem 5: Solve $\boldsymbol{\frac{1}{x} = \frac{6}{5x} + 1}$
# Explanation:
## Step1: Common denominator ($5x$)
Multiply all terms by $5x$:
$5x \cdot \frac{1}{x} = 5x \cdot \frac{6}{5x} + 5x \cdot 1$
Simplify: $5 = 6 + 5x$

## Step2: Solve for $x$
Subtract 6: $-1 = 5x$
Divide by 5: $x = -\frac{1}{5}$

## Step3: Check validity
Denominators: $x
eq 0$, $5x
eq 0$. $x = -\frac{1}{5}$ is valid.
# Answer: $x = -\frac{1}{5}$

### Problem 7: Solve $\boldsymbol{\frac{1}{v} + \frac{3v + 12}{v^2 - 5v} = \frac{7v - 56}{v^2 - 5v}}$
# Explanation:
## Step1: Factor denominators ($v^2 - 5v = v(v - 5)$)
Common denominator: $v(v - 5)$. Multiply all terms by $v(v - 5)$:
$v(v - 5) \cdot \frac{1}{v} + v(v - 5) \cdot \frac{3v + 12}{v(v - 5)} = v(v - 5) \cdot \frac{7v - 56}{v(v - 5)}$
Simplify: $v - 5 + 3v + 12 = 7v - 56$

## Step2: Combine like terms
$4v + 7 = 7v - 56$
Subtract $4v$: $7 = 3v - 56$
Add 56: $63 = 3v$
Divide by 3: $v = 21$

## Step3: Check validity
Denominators: $v
eq 0$, $v - 5
eq 0$ (so $v
eq 5$). $v = 21$ is valid.
# Answer: $v = 21$

### Problem 9: Solve $\boldsymbol{\frac{1}{n - 8} - 1 = \frac{7}{n - 8}}$
# Explanation:
## Step1: Common denominator ($n - 8$)
Multiply all terms by $n - 8$ ( $n
eq 8$ ):
$(n - 8) \cdot \frac{1}{n - 8} - (n - 8) \cdot 1 = (n - 8) \cdot \frac{7}{n - 8}$
Simplify: $1 - (n - 8) = 7$

## Step2: Solve for $n$
Expand: $1 - n + 8 = 7$
Combine terms: $9 - n = 7$
Subtract 9: $-n = -2$
Multiply by $-1$: $n = 2$

## Step3: Check validity
Denominator: $n - 8
eq 0 \implies n
eq 8$. $n = 2$ is valid.
# Answer: $n = 2$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Solve \(\boldsymbol{\frac{1}{6k^2} = \frac{1}{3k^2} - \frac{1}{k}}\)

Step1: Find common denominator (\(6k^2\))

Step2: Solve for \(k\)

Step3: Check for extraneous solutions

Step1: Common denominator (\(6b^2\))

Step2: Solve for \(b\)

Step3: Check validity

Step1: Common denominator (\(5x\))

Step2: Solve for \(x\)

Step3: Check validity

Answer:

Problem 3: Solve \(\boldsymbol{\frac{1}{6b^2} + \frac{1}{6b} = \frac{1}{b^2}}\)