Lily begins solving the equation 4(x – 1) – x = 3(x + 5) – 11. Her work is shown below.

Question

4(x – 1) – x = 3(x + 5) – 11
4x – 4 – x = 3x + 15 – 11
3x – 4 = 3x + 4
How can her partial solution be interpreted?

The equation has one solution: x = 0.
The equation has one solution: x = 8.
The equation has no solution.
The equation has infinite solutions.

Answer 1

Let's analyze Lily's work step by step to understand the implications of her partial solution.

Starting from her equation: \[ 4(x - 1) - x = 3(x + 5) - 11 \]

She correctly expanded both sides: \[ 4x - 4 - x = 3x + 15 - 11 \] which can be simplified to: \[ 4x - x - 4 = 3x + 15 - 11 \] \[ 3x - 4 = 3x + 4 \]

Next, when she reaches: \[ 3x - 4 = 3x + 4 \]

Notice that both sides have the term \( 3x \). If we subtract \( 3x \) from both sides, we are left with: \[ -4 = 4 \]

This is a contradiction, meaning that there are no values of \( x \) that satisfy the original equation. Therefore, the interpretation of her partial solution indicates that:

The equation has no solution.