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Solve the following equation. Then, state whether the equation is an identity, a conditional equation, or an inconsistent equation.

[tex]\[ \frac{3x}{x+2} - \frac{6}{x-2} = \frac{3x^2 + 12}{x^2 - 4} \][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is [tex]\(\square\)[/tex]. (Simplify your answer)
B. The solution set is [tex]\(\{ x \mid x \text{ is a real number} \}\)[/tex].
C. The solution set is [tex]\(\varnothing\)[/tex].

Is the equation an identity, a conditional equation, or an inconsistent equation?

1. Identity
2. Conditional equation
3. Inconsistent equation



Answer :

GinnyAnswer
Let's solve the given equation step-by-step and determine the type of the equation.

Given:
[tex]\[ \frac{3 x}{x+2} - \frac{6}{x-2} = \frac{3 x^2 + 12}{x^2 - 4} \][/tex]

Step 1: Simplify both sides of the equation.

First, simplify the left-hand side:
[tex]\[ \frac{3 x}{x+2} - \frac{6}{x-2} \][/tex]
We can combine these by expressing them over a common denominator, which is [tex]\((x + 2)(x - 2)\)[/tex]:
[tex]\[ \frac{3 x (x - 2) - 6 (x + 2)}{(x + 2)(x - 2)} \][/tex]
[tex]\[ = \frac{3 x^2 - 6x - 6x - 12}{x^2 - 4} \][/tex]
[tex]\[ = \frac{3 x^2 - 12x - 12}{x^2 - 4} \][/tex]
[tex]\[ = \frac{3 (x^2 - 4x - 4)}{x^2 - 4} \][/tex]

Now, rewrite the right-hand side:
[tex]\[ \frac{3 x^2 + 12}{x^2 - 4} \][/tex]
[tex]\[ = \frac{3 (x^2 + 4)}{x^2 - 4} \][/tex]

Step 2: Equate the simplified forms.

We now have:
[tex]\[ \frac{3 (x^2 - 4x - 4)}{x^2 - 4} = \frac{3 (x^2 + 4)}{x^2 - 4} \][/tex]

Since the denominators are the same, we equate the numerators:
[tex]\[ 3 (x^2 - 4x - 4) = 3 (x^2 + 4) \][/tex]

Step 3: Solve for [tex]\(x\)[/tex].

Divide both sides by 3:
[tex]\[ x^2 - 4x - 4 = x^2 + 4 \][/tex]

Subtract [tex]\(x^2\)[/tex] from both sides:
[tex]\[ -4x - 4 = 4 \][/tex]

Add 4 to both sides:
[tex]\[ -4x = 8 \][/tex]

Divide by -4:
[tex]\[ x = -2 \][/tex]

Step 4: Verify the solution.

Substitute [tex]\(x = -2\)[/tex] back into the original equation to check if it satisfies the equation:
[tex]\[ \frac{3 (-2)}{-2 + 2} - \frac{6}{-2 - 2} = \frac{3 (-2)^2 + 12}{(-2)^2 - 4} \][/tex]

Evaluate the left-hand side:
[tex]\[ \frac{3 (-2)}{0} - \frac{6}{-4} \quad (\text{undefined due to division by zero}) \][/tex]

Since [tex]\(x = -2\)[/tex] results in an undefined expression in the original equation, it is not a valid solution.

Step 5: Determine the type of equation.

Since there are no valid solutions, the solution set is empty.

Hence, the solution set is:
[tex]\[ \varnothing \][/tex]

And the type of the equation is:
[tex]\[ \text{Inconsistent equation} \][/tex]

The correct choices are:

C. [tex]\(\varnothing\)[/tex]

Inconsistent equation

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