Let's evaluate the given inequality step by step and identify the student's mistakes.
The original inequality is:
[tex]\[ 31 < -5x + 6 \][/tex]
### Step 1: Subtract 6 from Both Sides
To isolate the term involving [tex]\(x\)[/tex], we first subtract 6 from both sides:
[tex]\[ 31 - 6 < -5x + 6 - 6 \][/tex]
This simplifies to:
[tex]\[ 25 < -5x \][/tex]
So far, the student performed this step correctly.
### Step 2: Solving for [tex]\(x\)[/tex]
Next, we need to solve for [tex]\(x\)[/tex] by dividing both sides of the inequality by [tex]\(-5\)[/tex]. When dividing by a negative number, the direction of the inequality sign must be flipped.
[tex]\[ \frac{25}{-5} > x \][/tex]
Dividing 25 by [tex]\(-5\)[/tex] yields:
[tex]\[ -5 > x \][/tex]
This means:
[tex]\[ x < -5 \][/tex]
### Mistakes Identified:
1. Direction of Inequality Sign: While the student correctly identified the need to divide by [tex]\(-5\)[/tex], they did not flip the inequality sign. The inequality should be [tex]\( x < -5 \)[/tex] instead of [tex]\( -5 < x \)[/tex].
2. Incorrect Final Inequality: The student's final answer was incorrect because they failed to switch the direction of the inequality sign, resulting in the wrong final inequality [tex]\( -5 < x \)[/tex].
### Conclusion:
The correct final inequality after solving [tex]\( 31 < -5x + 6 \)[/tex] should be:
[tex]\[ x < -5 \][/tex]
