Adam solved this equation and identified the number of solutions.

Question

24x – 22 = 4(6x – 1)

24x – 22 = 24x – 4

24x = 24x + 18

0 = 18

The equation has infinitely many solutions.

When Adam verified his answer, it didn’t work. What was his mistake?

He used the distributive property incorrectly in the first step.
He used the addition property of equality incorrectly in the second step.
He should have found that the equation has one solution of x = 18.
He should have found that there are no solutions because the statement is false.

Answer 1

Let's analyze the solution step by step to identify Adam's mistake.

Starting from the original equation:

\[ 24x - 22 = 4(6x - 1) \]

Step 1: Apply the distributive property on the right side.

\[ 4(6x - 1) = 24x - 4 \]

So we have:

\[ 24x - 22 = 24x - 4 \]

Step 2: Now, subtract \( 24x \) from both sides:

\[ 24x - 22 - 24x = 24x - 4 - 24x \]

This simplifies to:

\[ -22 = -4 \]

Step 3: Then, adding 22 to both sides gives:

\[ 0 = 18 \]

This statement is false.

From this analysis, Adam concluded the equation has infinitely many solutions, but that is incorrect. The derived statement \( 0 = 18 \) indicates a contradiction, meaning there are no solutions to the equation.

Thus, Adam's mistake is that he should have found that there are no solutions because the statement is false.

Correct choice: He should have found that there are no solutions because the statement is false.