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

$\begin{array}{c}

4(x-1)-x=3(x+5)-11 \\

4 x-4-x=3 x+15-11 \\

3 x-4=3 x+4

\end{array}$

How can her partial solution be interpreted?

AThe equation has one solution: $x=0$.
BThe equation has one solution: $x=8$.
CThe equation has no solution.
DThe equation has infinite solutions.

Question

03-11-2026
Mathematics

Answered

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

$\begin{array}{c}

4(x-1)-x=3(x+5)-11 \\

4 x-4-x=3 x+15-11 \\

3 x-4=3 x+4

\end{array}$

How can her partial solution be interpreted?

AThe equation has one solution: $x=0$.
BThe equation has one solution: $x=8$.
CThe equation has no solution.
DThe equation has infinite solutions.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T17:31:01-05:00

Start with the simplified equation: $3x-4 = 3x+4$.
Attempt to isolate $x$ by subtracting $3x$ from both sides, resulting in $-4 = 4$.
Recognize that $-4 = 4$ is a contradiction, which means no value of $x$ can satisfy the equation.
Conclude that the equation has no solution.
Final Answer: $\boxed{\text{The equation has no solution.}}$

Explanation

Analysis of the Partial Solution

We are given the equation $4(x-1)-x = 3(x+5)-11$. Lily simplified it to $3x-4 = 3x+4$. To understand the implications of this result, we need to analyze the equality $3x-4 = 3x+4$ further. An equation is a statement that two expressions are equal; solving it involves finding the value(s) of the variable $x$ that make the statement true. If the manipulation of the equation leads to a mathematical contradiction—an impossible statement—then there are no values of $x$ that satisfy the original equation.

Interpreting the Contradiction

To isolate the variable $x$, we can subtract $3x$ from both sides of the equation $3x-4 = 3x+4$:
\[3x - 3x - 4 = 3x - 3x + 4\]
This simplifies to:
\[-4 = 4\]
The statement $-4 = 4$ is clearly false, as $-4$ does not equal $4$. This is a logical contradiction. Since our manipulations are equivalent, this contradiction proves that the original equation can never be true for any real number $x$. Therefore, the equation has no solution.

Conclusion

Given the options provided, the result of our analysis that the equation leads to a contradiction ($-4=4$) directly indicates that there is no value of $x$ that satisfies the equation. Thus, the correct interpretation is that the equation has no solution.

Examples

In real-world applications, identifying that an equation has 'no solution' is crucial in system modeling and logistics. For example, if you were trying to schedule two delivery trucks with conflicting constraints—such as requiring a truck to be in two different cities at the same time—the mathematical model of your schedule would lead to a contradiction, similar to the one we just solved. Recognizing this 'no solution' state prevents you from attempting the impossible and prompts you to adjust your constraints or rethink your logistics plan.