The only solution to the equation is 1

Question

Question 3
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

The equation solved correctly would show that it has
solution(s).

Question 4
4. Select the equation that has infinitely many solutions.(1 point)

Answer 1

Let's analyze the steps provided by the student to check the work and answer the questions.

1. Check the student's steps:

   • Starting equation: \( 4(2 - 3x) = x - 2(2x + 1) \)

   • Expanding both sides: \[ 8 - 12x = x - 4x - 2 \] Simplifying the right side: \[ 8 - 12x = -3x - 2 \]

   • Rearranging gives: \[ 8 - 12x = -3x - 2 \]

   • Adding \( 12x \) and \( 3x \) to both sides: \[ 8 = 9x - 2 \]

   • Adding 2 to both sides: \[ 10 = 9x \]

   • Dividing by 9 gives: \[ x = \frac{10}{9} \]

Thus, the equation has a unique solution, \( x = \frac{10}{9} \), not "No solution."

Answer to Question 3

The equation solved correctly would show that it has 1 solution.

Question 4

Equation with infinitely many solutions: An equation has infinitely many solutions if it simplifies to an identity, such as:

\[ 0 = 0 \]

For example: \[ 2x + 3 = 2(x + \frac{3}{2}) \]

This simplifies to the same expression on both sides.

To select from multiple-choice options (if provided), look for an equation where both sides can be manipulated to yield a tautology like \( 0 = 0 \).

Answer 2

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 or correctly

The equation solved correctly would show that it has
solution(s).
one or two or infinitely many or three

Answer 3

Let's closely analyze the steps taken by the student to determine whether their solution is correct or incorrect:

1. Original Equation: \[ 4(2−3x) = x−2(2x+1) \]

2. Distributing Both Sides: \[ 8 - 12x = x - (4x + 2) \] (Here, \( -2(2x + 1) \) expands to \( -4x - 2 \))

   This simplifies to: \[ 8 - 12x = x - 4x - 2 \]

3. Combining Terms: Simplifying the right side: \[ 8 - 12x = -3x - 2 \]

4. Rearranging the Equation: By adding \( 12x \) and \( 3x \) to both sides, we have: \[ 8 = 9x - 2 \] We then add \( 2 \) to both sides: \[ 10 = 9x \]

5. Solving for \( x \): Finally, we divide by \( 9 \): \[ x = \frac{10}{9} \]

Conclusion

The student's interpretation of "No solution" appears to be incorrect. The step that led to "No solution" is not a valid conclusion based on the preceding calculations. Instead, the correct analysis shows that the equation has:

• One solution: \( x = \frac{10}{9} \)

Final Answers

• The equation was solved incorrectly.
• The equation solved correctly would show that it has one solution.