Question

4. Select the equation that has infinitely many solutions.(1 point) Responses 3x=2x+1 3 x is equal to 2 x plus 1 3x+5=3x−5 3 x plus 5 is equal to 3 x minus 5 4x+1=4(x+1) 4 x plus 1 is equal to 4 times open paren x plus 1 close paren 3−3x=−3(x−1)

Answer 1

3. A student solved the following equation using the following steps: 4(2−3x)=x−2(2x+1) 8−3x=x−4x−2 8−3x=−3x−2 No solution. (2 points) Based on the student's work, the equation was solved incorrectly The equation solved correctly would show that it has one solution(s).

Answer 2

5. A student solved the following equation using the following steps: 2(2x+2)=4(x+4) 4x+4=4x+4 4=4 Infinitely many solutions (2 points) Based on the student's work, the equation was solved The equation solved correctly would show that it has solution(s).

Answer 3

To determine which equation has infinitely many solutions, we need to look for an equation that simplifies to a true statement for all values of \(x\).

1. **Equation: \(3x = 2x + 1\)**
Solving for \(x\):
\[3x - 2x = 1\]
\[x = 1\]
*This has a single solution.*

2. **Equation: \(3x + 5 = 3x - 5\)**
Solving for \(x\):
\[3x + 5 - 3x = -5\]
\[5 = -5\]
*This is false and has no solutions.*

3. **Equation: \(4x + 1 = 4(x + 1)\)**
Expanding the right-hand side:
\[4x + 1 = 4x + 4\]
\[1 = 4\]
*This is false and has no solutions.*

4. **Equation: \(3 - 3x = -3(x - 1)\)**
Expanding the right-hand side:
\[3 - 3x = -3x + 3\]
\[3 - 3x + 3x = 3\]
\[3 = 3\]
*This is a true statement for all values of \(x\), meaning it has infinitely many solutions.*

Therefore, the equation that has infinitely many solutions is:

**\(3 - 3x = -3(x - 1)\)**.

Answer 4

Let's analyze the student's work step-by-step to determine if the solution is correct and assess why they reached the conclusion of "no solution."

The original equation given is:

\[
4(2 - 3x) = x - 2(2x + 1)
\]

Now, let's go through the steps to solve this equation correctly.

1. **Distribute on both sides:**
\[
4(2) - 4(3x) = x - 2(2x) - 2(1)
\]
This simplifies to:
\[
8 - 12x = x - 4x - 2
\]

2. **Combine like terms on the right side:**
\[
8 - 12x = x - 4x - 2
\]
This simplifies further to:
\[
8 - 12x = -3x - 2
\]

3. **Rearranging terms:**
To isolate \(x\), we can add \(12x\) to both sides:
\[
8 = -3x + 12x - 2
\]
\[
8 = 9x - 2
\]

4. **Add \(2\) to both sides:**
\[
8 + 2 = 9x
\]
\[
10 = 9x
\]

5. **Divide by \(9\):**
\[
x = \frac{10}{9}
\]

This indicates that the equation has **one solution**: \(x = \frac{10}{9}\).

Now, check the student's conclusion: if they stated "no solution" after the step \(8 - 3x = -3x - 2\), this is incorrect.

At this point in the problem, the two sides can be simplified further and lead to a single solution. Thus, **the correct statement would be that the equation, when solved correctly, shows that it has one solution.**

### Conclusion:
- **The student's work shows that the equation was solved incorrectly**.
- The **correct solution would indicate that it has one solution**.

Answer 5

Let's review the student's work based on the equation given and determine if their conclusion about the number of solutions is correct.

The original equation is:

\[
2(2x + 2) = 4(x + 4)
\]

When we step through the student's solution:

1. **Distribute on both sides:**
\[
4x + 4 = 4x + 16
\]

2. **Now, the student incorrectly wrote:**
\[
4x + 4 = 4x + 4
\]
instead of writing \(4x + 4 = 4x + 16\).

That’s an important mistake because if we had correctly continued from that step, we would have:

3. **Subtract \(4x\) from both sides:**
\[
4 = 16
\]

4. **This statement (4 = 16) is false, which indicates that there are no solutions for the equation.**

### Conclusion:
- The student's conclusion that there are **infinitely many solutions** is incorrect because the steps leading to \(4 = 16\) reveal that there are **no solutions** at all.
- Thus, the **equation solved correctly would show that it has no solutions**, not infinitely many.

So, to finalize:
- **The statement on the number of solutions is incorrect. The correct conclusion is that the equation has no solutions.**