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:
-
\(2(x + 4) - x\)
- This simplifies to \(2x + 8 - x = x + 8\).
-
\(x + 8\)
- This is equal to \(x + 8\).
-
\(4(x + 2) - 5x\)
- This simplifies to \(4x + 8 - 5x = -x + 8\).
-
\(x\)
- 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:
\[ \boxed{4(x + 2) - 5x} \]
1 answer