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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)

5 answers

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:

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

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).

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.
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).

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:

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

  2. 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.