Question
Given: -2(x - 4) = 2x + 12 Prove: x = -1
1s -2(x - 4) = 2x + 12 Prove: x = -1. 1r _____. 2s _____. 2r distributive property. 3s _____. 3r _____. 4s -4x=4. 4r _____. 5s _____. 5r division property of equality.
1s -2(x - 4) = 2x + 12 Prove: x = -1. 1r _____. 2s _____. 2r distributive property. 3s _____. 3r _____. 4s -4x=4. 4r _____. 5s _____. 5r division property of equality.
Answers
GPT-4o mini
Let's go through the steps to solve the equation \(-2(x - 4) = 2x + 12\) and establish the proof that \(x = -1\).
### Proof Steps:
1. Start with the original equation:
\(-2(x - 4) = 2x + 12\)
**(1s)**: \(-2(x - 4) = 2x + 12\)
**(1r)**: Given.
2. Apply the distributive property on the left side:
\(-2 \cdot x + -2 \cdot (-4) = 2x + 12\)
So we get:
\(-2x + 8 = 2x + 12\)
**(2s)**: \(-2x + 8 = 2x + 12\)
**(2r)**: Distributive Property.
3. Rearrange the equation to isolate \(x\):
Start by adding \(2x\) to both sides:
\(8 = 2x + 2x + 12\)
This simplifies to:
\(8 = 4x + 12\)
**(3s)**: \(8 = 4x + 12\)
**(3r)**: Addition Property of Equality.
4. Now, subtract \(12\) from both sides:
\(8 - 12 = 4x\)
This gives us:
\(-4 = 4x\)
**(4s)**: \(-4 = 4x\)
**(4r)**: Subtraction Property of Equality.
5. Finally, divide both sides by \(4\) to solve for \(x\):
\(-4/4 = x\)
This simplifies to:
\(x = -1\)
**(5s)**: \(x = -1\)
**(5r)**: Division Property of Equality.
### Conclusion:
From the proof above, we have demonstrated that starting from the equation \(-2(x - 4) = 2x + 12\), through the appropriate steps including the use of the distributive property, addition, subtraction, and division properties of equality, we have successfully proven that \(x = -1\).
### Proof Steps:
1. Start with the original equation:
\(-2(x - 4) = 2x + 12\)
**(1s)**: \(-2(x - 4) = 2x + 12\)
**(1r)**: Given.
2. Apply the distributive property on the left side:
\(-2 \cdot x + -2 \cdot (-4) = 2x + 12\)
So we get:
\(-2x + 8 = 2x + 12\)
**(2s)**: \(-2x + 8 = 2x + 12\)
**(2r)**: Distributive Property.
3. Rearrange the equation to isolate \(x\):
Start by adding \(2x\) to both sides:
\(8 = 2x + 2x + 12\)
This simplifies to:
\(8 = 4x + 12\)
**(3s)**: \(8 = 4x + 12\)
**(3r)**: Addition Property of Equality.
4. Now, subtract \(12\) from both sides:
\(8 - 12 = 4x\)
This gives us:
\(-4 = 4x\)
**(4s)**: \(-4 = 4x\)
**(4r)**: Subtraction Property of Equality.
5. Finally, divide both sides by \(4\) to solve for \(x\):
\(-4/4 = x\)
This simplifies to:
\(x = -1\)
**(5s)**: \(x = -1\)
**(5r)**: Division Property of Equality.
### Conclusion:
From the proof above, we have demonstrated that starting from the equation \(-2(x - 4) = 2x + 12\), through the appropriate steps including the use of the distributive property, addition, subtraction, and division properties of equality, we have successfully proven that \(x = -1\).
3s is supposed to be -4x+8=12
GPT-4o mini
Thank you for the clarification. Let's adjust the proof accordingly to incorporate the step you provided. Here’s the proof step-by-step:
### Proof Steps:
1. Start with the original equation:
\(-2(x - 4) = 2x + 12\)
**(1s)**: \(-2(x - 4) = 2x + 12\)
**(1r)**: Given.
2. Apply the distributive property on the left side:
\(-2 \cdot x + -2 \cdot (-4) = 2x + 12\)
This gives us:
\(-2x + 8 = 2x + 12\)
**(2s)**: \(-2x + 8 = 2x + 12\)
**(2r)**: Distributive Property.
3. Rearranging the equation:
Start by subtracting \(2x\) from both sides:
\(-2x - 2x + 8 = 12\)
This simplifies to:
\(-4x + 8 = 12\)
**(3s)**: \(-4x + 8 = 12\)
**(3r)**: Subtraction Property of Equality.
4. Now, subtract \(8\) from both sides:
\(-4x + 8 - 8 = 12 - 8\)
This gives us:
\(-4x = 4\)
**(4s)**: \(-4x = 4\)
**(4r)**: Subtraction Property of Equality.
5. Finally, divide both sides by \(-4\) to solve for \(x\):
\(\frac{-4x}{-4} = \frac{4}{-4}\)
This simplifies to:
\(x = -1\)
**(5s)**: \(x = -1\)
**(5r)**: Division Property of Equality.
### Conclusion:
The steps above have been modified to match your guidance, and they still lead to the conclusion that \(x = -1\) from the original equation \(-2(x - 4) = 2x + 12\). Thank you for your patience!
### Proof Steps:
1. Start with the original equation:
\(-2(x - 4) = 2x + 12\)
**(1s)**: \(-2(x - 4) = 2x + 12\)
**(1r)**: Given.
2. Apply the distributive property on the left side:
\(-2 \cdot x + -2 \cdot (-4) = 2x + 12\)
This gives us:
\(-2x + 8 = 2x + 12\)
**(2s)**: \(-2x + 8 = 2x + 12\)
**(2r)**: Distributive Property.
3. Rearranging the equation:
Start by subtracting \(2x\) from both sides:
\(-2x - 2x + 8 = 12\)
This simplifies to:
\(-4x + 8 = 12\)
**(3s)**: \(-4x + 8 = 12\)
**(3r)**: Subtraction Property of Equality.
4. Now, subtract \(8\) from both sides:
\(-4x + 8 - 8 = 12 - 8\)
This gives us:
\(-4x = 4\)
**(4s)**: \(-4x = 4\)
**(4r)**: Subtraction Property of Equality.
5. Finally, divide both sides by \(-4\) to solve for \(x\):
\(\frac{-4x}{-4} = \frac{4}{-4}\)
This simplifies to:
\(x = -1\)
**(5s)**: \(x = -1\)
**(5r)**: Division Property of Equality.
### Conclusion:
The steps above have been modified to match your guidance, and they still lead to the conclusion that \(x = -1\) from the original equation \(-2(x - 4) = 2x + 12\). Thank you for your patience!