Classify each of the following equations. 12 ( x − 4 ) = 4 ( 3 x + 12 ) 12\left(x-4\right)=4\left(3x+12\right)

Welcome back. In this problem, we want to classify the equation 12 times x minus four equals four times three x plus 12. To classify the equation, we're going to need to figure out how many solutions this equation has, so let's start by solving the equation. We need to simplify both sides of the equation first. So let's distribute this 12 on the left side of the equation. We have 12 times x, which is 12 x, and then 12 times negative four, which is minus 48. On the right side of the equation, we have to distribute the four, so we have four times three x, which is 12 x, and then we have four times 12, which is 48. Next, we have to try to get all of our variable terms on one side of the equation and all of our constant terms on the other side of the equation. We currently have variables and constants on both sides of the equation, so let's try to get the variables to the left and the constants to the right, starting with the constants. We need to get rid of this minus 48 on the left side of the equation, which we can do by adding 48 to both sides of our equation. It cancels the minus 48 on the left, leaving us with 12 x. And on the right, we have 12 x plus 48 plus 48, which is plus 96. Next, we wanna get our variable terms to the left side of the equation. We need to get rid of this 12 x on the right side, which we can do by subtracting 12 x on both sides of the equation. That's going to cancel the 12 x on the right side of the equation, leaving us with just 96. But it's also going to cancel the 12 x on the left side of the equation, leaving us with zero. So we get the statement zero equals 96. And that's a false statement. That's always going to be false no matter what we try to plug in for x. So we have no solution, and we got a false statement, which is a contradiction. So we can say that this equation is a contradiction because it has no solution and results in a false statement. Great work.

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Multiple Choice

Classify each of the following equations.
$12 (x - 4) = 4 (3 x + 12)$

Conditional

Identity

Contradiction

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Verified step by step guidance

Start by expanding both sides of the equation \$12\left(x - 4\right) = 4\left(3x + 12\right)$ using the distributive property.

Distribute the constants: multiply 12 by each term inside the first parentheses and 4 by each term inside the second parentheses, resulting in \$12x - 48 = 12x + 48$.

Next, get all variable terms on one side and constants on the other by subtracting \$12x$ from both sides, which simplifies the equation to $-48 = 48$.

Analyze the simplified equation: since $-48 = 48$ is a false statement, it means there is no value of $x$ that can satisfy the original equation.

Conclude that the equation is a contradiction because it has no solution, unlike a conditional equation (which has one or more solutions) or an identity (which is true for all values of $x$).

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Classify each of the following equations.12(x−4)=4(3x+12)12\left(x-4\right)=4\left(3x+12\right)

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Classify each of the following equations.
$12 (x - 4) = 4 (3 x + 12)$