Question

Which equation has infinitely many solutions?
5(2x + 4) = 10x – 12
5(2x + 4) = 10(x + 2)
5(2x + 4) = 12x
5(2x + 10) = 20(x + 1)

Answer 1

To determine which of the given equations has infinitely many solutions, we can simplify each equation and see if they lead to a true statement (such as \(0 = 0\)) that is valid for all values of \(x\).

1. **Equation 1:** \(5(2x + 4) = 10x - 12\)

Simplifying the left side:
\[
5(2x + 4) = 10x + 20
\]
Setting the two sides equal gives:
\[
10x + 20 = 10x - 12
\]
Subtracting \(10x\) from both sides:
\[
20 = -12 \quad \text{(False)}
\]
This has no solutions.

2. **Equation 2:** \(5(2x + 4) = 10(x + 2)\)

Simplifying the left side:
\[
5(2x + 4) = 10x + 20
\]
Simplifying the right side:
\[
10(x + 2) = 10x + 20
\]
Setting the two sides equal gives:
\[
10x + 20 = 10x + 20
\]
This is a true statement, so this equation has infinitely many solutions.

3. **Equation 3:** \(5(2x + 4) = 12x\)

Simplifying the left side:
\[
5(2x + 4) = 10x + 20
\]
Setting the two sides equal gives:
\[
10x + 20 = 12x
\]
Rearranging gives:
\[
20 = 2x \quad \Rightarrow \quad x = 10
\]
This equation has one solution.

4. **Equation 4:** \(5(2x + 10) = 20(x + 1)\)

Simplifying the left side:
\[
5(2x + 10) = 10x + 50
\]
Simplifying the right side:
\[
20(x + 1) = 20x + 20
\]
Setting the two sides equal gives:
\[
10x + 50 = 20x + 20
\]
Rearranging gives:
\[
50 - 20 = 20x - 10x \quad \Rightarrow \quad 30 = 10x \quad \Rightarrow \quad x = 3
\]
This equation has one solution.

Now, to summarize:
- The first equation has no solutions.
- The second equation has infinitely many solutions.
- The third equation has one solution.
- The fourth equation has one solution.

**Conclusion: The equation with infinitely many solutions is:**
\[
5(2x + 4) = 10(x + 2)
\]