To find the student's error in solving the inequality, let's go through the solution step-by-step and identify where the mistake occurred.
Here is the inequality the student needed to solve:
[tex]\[ 31 < -5x + 6 \][/tex]
### Step 1: Isolate the term involving [tex]\( x \)[/tex]
The first step is to isolate the term involving [tex]\( x \)[/tex] on one side of the inequality. The student correctly subtracted 6 from both sides of the inequality, but this step was mentioned as incorrect. Let's clarify why it's actually correct:
[tex]\[ 31 - 6 < -5x + 6 - 6 \][/tex]
This simplifies to:
[tex]\[ 25 < -5x \][/tex]
### Step 2: Divide by the coefficient of [tex]\( x \)[/tex]
Next, the coefficient of [tex]\( x \)[/tex] is [tex]\(-5\)[/tex]. We divide both sides of the inequality by [tex]\(-5\)[/tex] to isolate [tex]\( x \)[/tex]. This step should be done cautiously because when we divide or multiply both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. Here's how it should be done correctly:
[tex]\[ \frac{25}{-5} > x \][/tex]
### Correction in the direction of the inequality
Reversing the direction of the inequality sign when dividing by a negative number, we get:
[tex]\[ -5 > x \][/tex]
or, equivalently,
[tex]\[ x < -5 \][/tex]
### Conclusion
The student's mistake was in the final step. Instead of switching the direction of the inequality sign after dividing by [tex]\(-5\)[/tex], the student kept it the same and wrote [tex]\(-5 < x\)[/tex], which is incorrect. The correct solution should be [tex]\( x < -5 \)[/tex].
So, the student's error was not switching the direction of the inequality sign after dividing by a negative number.
