Question

Solve for x to determine if the equation has one, none, or infinite solutions.
3(x+2)−6=3x
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Answer 1

Solve for x to determine if the equation has one, none, or infinite solutions.
4x=3(x+5)−15

Answer 2

Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)−7x=3x+6

Answer 3

To solve the equation \( 3(x + 2) - 6 = 3x \), we will first simplify the left side.

1. Distribute the \( 3 \) into \( (x + 2) \):
\[
3x + 6 - 6 = 3x
\]

2. Simplify the left side:
\[
3x + 6 - 6 = 3x \implies 3x = 3x
\]

3. Now, we have:
\[
3x = 3x
\]

This equation is always true, meaning it holds for any value of \( x \).

Since the equation is true for all values of \( x \), we conclude that the equation has **infinite solutions**.

Answer 4

To solve the equation \( 4x = 3(x + 5) - 15 \), we'll start by simplifying the right side.

1. Distribute the \( 3 \) on the right side:
\[
4x = 3x + 15 - 15
\]

2. Simplify the right side:
\[
4x = 3x + 0 \implies 4x = 3x
\]

3. Next, we can subtract \( 3x \) from both sides:
\[
4x - 3x = 0 \implies x = 0
\]

Now we have found a specific solution, \( x = 0 \).

Thus, there is **one solution** to the equation.

Answer 5

To solve the equation \( 5(2x + 2) - 7x = 3x + 6 \), we start by simplifying both sides.

1. Distribute the \( 5 \) on the left side:
\[
10x + 10 - 7x = 3x + 6
\]

2. Combine like terms on the left side:
\[
(10x - 7x) + 10 = 3x + 6 \implies 3x + 10 = 3x + 6
\]

3. Now we can isolate the terms involving \( x \):
\[
3x + 10 - 3x = 6 \implies 10 = 6
\]

The statement \( 10 = 6 \) is false, which means that there is no solution to the equation.

Therefore, the equation has **no solutions**.