Question

Solve for x.

3x + 2=x+8

A. 1.5

B. 2.5

• с.3

• D.5

Answer 1

Which equations have infinitely many solutions? Select TWO that apply.
A. 6(x-9) = 6x -54+x
B. 5(x - 8) + 10 = 5(x + 2)
C. 20 - 8x-6 = 2(7 - 4x)
D. 4(x+4) = 2(2x+5) + 6
E. 2x + 9 = 18 + 2x

Answer 2

To solve the equation \(3x + 2 = x + 8\), follow these steps:

1. Subtract \(x\) from both sides:

\[
3x - x + 2 = 8
\]

This simplifies to:

\[
2x + 2 = 8
\]

2. Next, subtract 2 from both sides:

\[
2x = 8 - 2
\]

This simplifies to:

\[
2x = 6
\]

3. Finally, divide both sides by 2:

\[
x = \frac{6}{2} = 3
\]

So the solution is:

\[
x = 3
\]

The correct answer is **C. 3**.

Answer 3

To determine which equations have infinitely many solutions, we need to simplify each equation and see if they result in a true statement (like \(0 = 0\)) or a false statement (like \(0 = 5\)). Equations that are the same on both sides will have infinitely many solutions. Let's analyze each option.

### A. \(6(x-9) = 6x - 54 + x\)

Distributing on the left side:

\[
6x - 54 = 6x - 54 + x
\]

Simplifying the right side:

\[
6x - 54 = 7x - 54
\]

Subtract \(6x\) from both sides:

\[
-54 = x - 54
\]

Add \(54\) to both sides:

\[
0 = x
\]

This has one solution (\(x = 0\)), not infinitely many.

---

### B. \(5(x - 8) + 10 = 5(x + 2)\)

Distributing on the left side:

\[
5x - 40 + 10 = 5x + 10
\]

This simplifies to:

\[
5x - 30 = 5x + 10
\]

Subtract \(5x\) from both sides:

\[
-30 = 10
\]

This is a false statement, so it has no solutions.

---

### C. \(20 - 8x - 6 = 2(7 - 4x)\)

Simplifying the left side:

\[
14 - 8x = 14 - 8x
\]

Both sides are identical, indicating this equation has infinitely many solutions.

---

### D. \(4(x+4) = 2(2x+5) + 6\)

Distributing:

\[
4x + 16 = 4x + 10 + 6
\]

Simplifying the right side:

\[
4x + 16 = 4x + 16
\]

Again, both sides are identical, indicating this equation also has infinitely many solutions.

---

### E. \(2x + 9 = 18 + 2x\)

Subtracting \(2x\) from both sides gives:

\[
9 = 18
\]

This is a false statement, so it has no solutions.

---

### Summary:
The equations with infinitely many solutions are:
- **C. \(20 - 8x - 6 = 2(7 - 4x)\)**
- **D. \(4(x+4) = 2(2x+5) + 6\)**

So the correct answers are **C** and **D**.