Question

1. sally solved the equation $\frac{3}{x-2} = \frac{x}{x-2}$ and got 2 and 3 as solutions. which of these reasonable and why?
2. solve the following rational equation:

$1 - \frac{4x}{x+4} = -\frac{5}{x+4}$

1. find the vertical and horizontal asymptotes, and use them to find the domain and range

$f(x) = \frac{1}{x+5} - 2$
domain: ____
range: ____

Explanation:

Problem 6

Step1: Identify undefined value

The denominator $x-2$ cannot be 0, so $x
eq 2$.

Step2: Solve the equation

Multiply both sides by $x-2$:
$3 = x$

Step3: Verify solutions

$x=2$ makes the original equation undefined, so it is an extraneous solution. $x=3$ is valid.

Step1: Identify undefined value

Denominator $x+4
eq 0$, so $x
eq -4$.

Step2: Multiply by common denominator

Multiply all terms by $x+4$:
$1 \cdot (x+4) - 4x = -5$

Step3: Simplify and solve

$x+4-4x = -5$
$-3x + 4 = -5$
$-3x = -9$
$x = 3$

Step4: Verify solution

$x=3$ does not make $x+4=0$, so it is valid.

Step1: Find vertical asymptote

Set denominator $x+5=0$, so $x=-5$.

Step2: Find horizontal asymptote

As $x \to \pm\infty$, $\frac{1}{x+5} \to 0$, so $y=0-2=-2$.

Step3: Determine domain

All real numbers except vertical asymptote: $x
eq -5$.

Step4: Determine range

All real numbers except horizontal asymptote: $y
eq -2$.

Answer:

Only 3 is reasonable. $x=2$ makes the denominator of the original rational equation equal to 0, so it is not in the domain and is an extraneous solution. $x=3$ satisfies the original equation.

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Problem 7