Solve the equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. (4x)/(x-3)=(12)/(x-3)+7
The Correct Answer and Explanation is:
Let’s solve the equation step by step:
The given equation is:4xx−3=12x−3+7\frac{4x}{x – 3} = \frac{12}{x – 3} + 7x−34x=x−312+7
Step 1: Eliminate the denominator
We notice that both terms on the left and right side of the equation have a denominator of x−3x – 3x−3. To simplify, we can multiply the entire equation by x−3x – 3x−3 (assuming x≠3x \neq 3x=3 because division by zero is undefined):(x−3)×4xx−3=(x−3)×(12x−3+7)(x – 3) \times \frac{4x}{x – 3} = (x – 3) \times \left(\frac{12}{x – 3} + 7\right)(x−3)×x−34x=(x−3)×(x−312+7)
This simplifies to:4x=12+7(x−3)4x = 12 + 7(x – 3)4x=12+7(x−3)
Step 2: Expand the terms
Now expand the right side:4x=12+7x−214x = 12 + 7x – 214x=12+7x−21
Simplify further:4x=7x−94x = 7x – 94x=7x−9
Step 3: Move terms involving xxx to one side
To isolate xxx, subtract 7x7x7x from both sides:4x−7x=−94x – 7x = -94x−7x=−9
Simplify:−3x=−9-3x = -9−3x=−9
Step 4: Solve for xxx
Now, divide both sides by −3-3−3:x=3x = 3x=3
Step 5: Check for restrictions
Recall that we excluded x=3x = 3x=3 earlier, since it would make the denominators in the original equation zero. Therefore, the solution x=3x = 3x=3 is not valid.
Conclusion
Since x=3x = 3x=3 leads to division by zero in the original equation, there is no valid solution.
This makes the equation inconsistent, meaning there is no solution for xxx.
