Find the student's error in solving the following inequality.

[tex]\[
\begin{array}{l}
31 \ \textless \ -5x + 6 \\
25 \ \textless \ -5x \\
-5 \ \textless \ x
\end{array}
\][/tex]

A. The student should have added 6 to both sides instead of subtracting it.
B. The student divided 25 by -5 incorrectly.
C. The student should have switched the direction of the inequality sign to get [tex]\(-5 \ \textgreater \ x\)[/tex] for a final answer.



Answer :

Certainly! Let's go through the solution step-by-step to identify where the student made errors and then solve the inequality correctly.

### Given Inequality:
[tex]\[ 31 < -5x + 6 \][/tex]

### Steps and Corrections:

1. Subtract 6 from both sides:

The student's first step was to subtract 6, but it appears there was an error in their process. Let's do it correctly:
[tex]\[ 31 - 6 < -5x + 6 - 6 \][/tex]
[tex]\[ 25 < -5x \][/tex]

2. Divide both sides by -5:

When dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.
[tex]\[ 25 < -5x \][/tex]
[tex]\[ \frac{25}{-5} > x \][/tex]
[tex]\[ -5 > x \][/tex]

3. Rewriting the solution properly:

The correct way to express this result is:
[tex]\[ x < -5 \][/tex]

Therefore, the final solution to the inequality [tex]\(31 < -5x + 6\)[/tex] is:
[tex]\[ x < -5 \][/tex]

### Identifying the Errors:

1. The student correctly identified that they need to subtract 6 from both sides but made an incorrect simplification following this step.

2. During the division step, the student failed to reverse the inequality when dividing by a negative number, leading them to [tex]\( -5 < x \)[/tex] instead of the correct [tex]\( x < -5 \)[/tex].

Therefore, the correct final solution is [tex]\( x < -5 \)[/tex].