Question

The expression on the left side of an equation is shown below.

Negative 4 (x minus 2) + 5 x = box

If the equation has no solution, which expression can be written in the box on the other side of the equation?
2(x + 4) – x
x + 8
4(x + 2) – 5x
x

Answer 1

Let's start by simplifying the left side of the equation:

Given:
\[
-4(x - 2) + 5x
\]

Distributing \(-4\):
\[
-4x + 8 + 5x
\]

Combining like terms:
\[
(-4x + 5x) + 8 = x + 8
\]

So, the left side simplifies to:
\[
x + 8 = \text{box}
\]

For the equation to have no solution, the right side (box) must be a constant that is not equal to \(x + 8\).

Now let's evaluate the provided expressions to see if any can produce a contradiction:

1. **\(2(x + 4) - x\)**
- This simplifies to \(2x + 8 - x = x + 8\).

2. **\(x + 8\)**
- This is equal to \(x + 8\).

3. **\(4(x + 2) - 5x\)**
- This simplifies to \(4x + 8 - 5x = -x + 8\).

4. **\(x\)**
- This is just \(x\).

The condition for no solution is when both sides of the equation can never be equal.

The only expression that does not equal \(x + 8\) (which is the simplified left side) is:
\(-x + 8\).

So **\(4(x + 2) - 5x\)** would lead to \(8 - x\), which is \(8\) when \(x = 0\) and does not equal \(x + 8\) when \(x\) takes on other values.

Thus, the expression that can be written in the box to ensure the equation has no solution is:

\[
\boxed{4(x + 2) - 5x}
\]