Question
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Consider the unfinished equation 12(x - 3) + 18 = _______. Match the following expressions with the number of solutions the equation would have with that expression on the right-hand side.
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
6(2x-3)
4(3x-3)
4(2x-3)
Consider the unfinished equation 12(x - 3) + 18 = _______. Match the following expressions with the number of solutions the equation would have with that expression on the right-hand side.
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
6(2x-3)
4(3x-3)
4(2x-3)
Answers
GPT-4o mini
To analyze the number of solutions for the equation \(12(x - 3) + 18 = \text{RHS}\) for different expressions on the right-hand side, we first simplify the left-hand side of the equation:
\[
12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18
\]
Now, we will evaluate each expression provided on the right-hand side:
1. **For \(6(2x - 3)\)**:
\[
12x - 18 = 6(2x - 3)
\]
Simplifying the right side:
\[
6(2x - 3) = 12x - 18
\]
This results in a true statement, which means there are infinitely many solutions because both sides are equal for all \(x\).
2. **For \(4(3x - 3)\)**:
\[
12x - 18 = 4(3x - 3)
\]
Simplifying the right side:
\[
4(3x - 3) = 12x - 12
\]
Setting the two sides equal:
\[
12x - 18 = 12x - 12
\]
Subtracting \(12x\) from both sides gives:
\[
-18 = -12
\]
This is a contradiction, meaning there are no solutions.
3. **For \(4(2x - 3)\)**:
\[
12x - 18 = 4(2x - 3)
\]
Simplifying the right side:
\[
4(2x - 3) = 8x - 12
\]
Setting the two sides equal:
\[
12x - 18 = 8x - 12
\]
Subtracting \(8x\) from both sides gives:
\[
4x - 18 = -12
\]
Adding \(18\) to both sides gives:
\[
4x = 6 \quad \Rightarrow \quad x = \frac{3}{2}
\]
This shows that there is one solution.
Now we can summarize the findings:
- \(6(2x-3)\): Infinitely many solutions
- \(4(3x-3)\): No solutions
- \(4(2x-3)\): One solution
Match the expressions with the number of solutions:
- \(6(2x-3)\): Infinitely many solutions
- \(4(3x-3)\): No solutions
- \(4(2x-3)\): One solution
Input the corresponding responses based on this analysis.
\[
12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18
\]
Now, we will evaluate each expression provided on the right-hand side:
1. **For \(6(2x - 3)\)**:
\[
12x - 18 = 6(2x - 3)
\]
Simplifying the right side:
\[
6(2x - 3) = 12x - 18
\]
This results in a true statement, which means there are infinitely many solutions because both sides are equal for all \(x\).
2. **For \(4(3x - 3)\)**:
\[
12x - 18 = 4(3x - 3)
\]
Simplifying the right side:
\[
4(3x - 3) = 12x - 12
\]
Setting the two sides equal:
\[
12x - 18 = 12x - 12
\]
Subtracting \(12x\) from both sides gives:
\[
-18 = -12
\]
This is a contradiction, meaning there are no solutions.
3. **For \(4(2x - 3)\)**:
\[
12x - 18 = 4(2x - 3)
\]
Simplifying the right side:
\[
4(2x - 3) = 8x - 12
\]
Setting the two sides equal:
\[
12x - 18 = 8x - 12
\]
Subtracting \(8x\) from both sides gives:
\[
4x - 18 = -12
\]
Adding \(18\) to both sides gives:
\[
4x = 6 \quad \Rightarrow \quad x = \frac{3}{2}
\]
This shows that there is one solution.
Now we can summarize the findings:
- \(6(2x-3)\): Infinitely many solutions
- \(4(3x-3)\): No solutions
- \(4(2x-3)\): One solution
Match the expressions with the number of solutions:
- \(6(2x-3)\): Infinitely many solutions
- \(4(3x-3)\): No solutions
- \(4(2x-3)\): One solution
Input the corresponding responses based on this analysis.