Question

4. Select the equation that has infinitely many solutions.(1 point)Responses3−3x=−3(x−1)3 minus 3 x is equal to negative 3 times open paren x minus 1 close paren3x=2x+13 x is equal to 2 x plus 13x+5=3x−53 x plus 5 is equal to 3 x minus 54x+1=4(x+1)4 x plus 1 is equal to 4 times open paren x plus 1 close parenQuestion 5

Answer 1

To determine which of the equations has infinitely many solutions, we need to look for an identity, which is an equation that is true for all values of \( x \).

Let's analyze each equation:

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

Expanding the right side:
\[
3 - 3x = -3x + 3
\]
Rearranging:
\[
3 - 3x + 3x = 3 \quad \text{(true for any } x\text{)}
\]
Thus, this equation is an identity and has infinitely many solutions.

2. **Equation 2: \( 3x = 2x + 1 \)**

Rearranging gives:
\[
3x - 2x = 1 \implies x = 1
\]
This equation has a single solution.

3. **Equation 3: \( 3x + 5 = 3x - 5 \)**

Rearranging gives:
\[
5 = -5 \quad \text{(not true)}
\]
This equation has no solutions.

4. **Equation 4: \( 4x + 1 = 4(x + 1) \)**

Expanding the right side:
\[
4x + 1 = 4x + 4
\]
Rearranging gives:
\[
1 = 4 \quad \text{(not true)}
\]
This equation has no solutions.

Based on this analysis, the equation with infinitely many solutions is:

**Answer: \( 3 - 3x = -3(x - 1) \)**