Question

Type your answer in the boxes for the correct statement or reasons for the following proof. Given: -2(x - 4) = 2x + 12 Prove: x = -1
Statement (1s): -2(x-4)= 2x+12

Reason (1r): _____.

Statement (2s): _____

Reason (2r): Distributive property.

Statement (3s): _____

Reason (3r): _____

Statement (4s): -4x=4

Reason (4r): _____

Statement (5s): _____

Reason (5r): division property of equality

Answer 1

What property is reason 4?

Answer 2

What property is reason 3?

Answer 3

Let's fill in the statements and reasons step-by-step for the proof of \( x = -1 \) given the equation \( -2(x - 4) = 2x + 12 \):

1. **Statement (1s)**: \(-2(x - 4) = 2x + 12\)
**Reason (1r)**: Given.

2. **Statement (2s)**: \(-2x + 8 = 2x + 12\)
**Reason (2r)**: Distributive property.

3. **Statement (3s)**: \(-2x - 2x + 8 = 12\)
**Reason (3r)**: Subtracting \(2x\) from both sides (combining like terms).

4. **Statement (4s)**: \(-4x + 8 = 12\)
**Reason (4r)**: Combining like terms.

5. **Statement (5s)**: \(-4x = 4\)
**Reason (5r)**: Subtracting 8 from both sides.

6. **Statement (6s)**: \(x = -1\)
**Reason (6r)**: Division property of equality (dividing both sides by -4).

So, the proof is:

1. Statement (1s): \(-2(x - 4) = 2x + 12\)
Reason (1r): Given.

2. Statement (2s): \(-2x + 8 = 2x + 12\)
Reason (2r): Distributive property.

3. Statement (3s): \(-4x + 8 = 12\)
Reason (3r): Adding \(2x\) to both sides (combining like terms).

4. Statement (4s): \(-4x = 4\)
Reason (4r): Subtracting 8 from both sides.

5. Statement (5s): \(x = -1\)
Reason (5r): Division property of equality (dividing both sides by -4).

Answer 4

The reasoning used in statement (4) is an application of the **subtraction property of equality**. Here, you are subtracting 8 from both sides of the equation:

From statement (3s):
\(-4x + 8 = 12\)

After applying the subtraction property of equality by subtracting 8 from both sides, you arrive at:
\(-4x = 4\)

So to clarify:

**Reason (4r)**: Subtraction property of equality.

Answer 5

In the context of statement (3), which states:

**Statement (3s)**: \(-2x - 2x + 8 = 12\)

The reasoning behind this step involves combining like terms. Specifically, you are combining the terms involving \(x\):

From statement (2s):
\(-2x + 8 = 2x + 12\)

When you bring the \(-2x\) to the left side, you rewrite it as:

\(-2x - 2x + 8 = 12\)

The specific property used here is the **combining like terms** or **associative property of addition**.

Thus, for your proof:

**Reason (3r)**: Combining like terms (though you could also mention the associative property of addition).