Solve for x to determine if the equation has one, none, or infinite solutions.

Question

6 (2x + 4) − 24 = 12x

A). The equation has one solution: x=0

B). The equation has no solution.

C). The equation has infinite solutions.

D). The equation has one solution: x=−20
.

Answer 1

To solve the equation \( 6(2x + 4) - 24 = 12x \), we start by simplifying and solving for \( x \).

1. Distribute \( 6 \) on the left side: \[ 12x + 24 - 24 = 12x \]

2. The equation simplifies to: \[ 12x = 12x \]

3. Next, we can see that both sides of the equation are equal. This indicates that the equation is true for all values of \( x \).

Since the equation holds for any \( x \), it has infinite solutions.

Thus, the answer is:

C). The equation has infinite solutions.

Answer 2

Solve for x to determine if the equation has one, none, or infinite solutions.

9 (x + 1) = 3 (5x + 1) − 12

A). The equation has one solution: x=0

B). The equation has infinite solutions.

C). The equation has no solution.

D). The equation has one solution: x=3
.

Answer 3

Let's solve the equation \( 9(x + 1) = 3(5x + 1) - 12 \) step by step.

1. Distribute on both sides:

   Left side: \[ 9(x + 1) = 9x + 9 \]

   Right side: \[ 3(5x + 1) - 12 = 15x + 3 - 12 = 15x - 9 \]

   Now we rewrite the equation: \[ 9x + 9 = 15x - 9 \]

2. Next, let's move all terms involving \( x \) to one side and constant terms to the other side. Subtract \( 9x \) from both sides: \[ 9 = 15x - 9x - 9 \]

   This simplifies to: \[ 9 = 6x - 9 \]

3. Now, add \( 9 \) to both sides: \[ 9 + 9 = 6x \] \[ 18 = 6x \]

4. Now, divide by 6: \[ x = 3 \]

Since we found one specific solution, we can conclude:

A). The equation has one solution: x=3.