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].
