Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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
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
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.