Certainly! Let's go through the solution step-by-step:
1. Initial Inequality:
[tex]\[
31 < -5x + 6
\][/tex]
2. Isolate the term with [tex]\( x \)[/tex]:
To isolate the term with [tex]\( x \)[/tex], we need to remove the constant term on the right-hand side. We do this by subtracting 6 from both sides:
[tex]\[
31 - 6 < -5x + 6 - 6
\][/tex]
Simplifying this, we get:
[tex]\[
25 < -5x
\][/tex]
3. Correcting the Mistake:
The student correctly isolated the term with [tex]\( x \)[/tex] up to this point. However, the mistake occurs in the next step.
4. Dividing by -5:
When we divide or multiply both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. Therefore, we divide both sides by -5 and switch the inequality sign:
[tex]\[
\frac{25}{-5} > x
\][/tex]
Simplifying this gives:
[tex]\[
-5 > x
\][/tex]
5. Rewriting the Final Step:
The inequality [tex]\( -5 > x \)[/tex] can also be written as:
[tex]\[
x < -5
\][/tex]
### Summary of Errors:
- The student did not switch the direction of the inequality sign when dividing by a negative number. The correct division resulted in [tex]\( -5 > x \)[/tex], and should have been reported as [tex]\( x < -5 \)[/tex].
- The initial steps were correct until the division by -5, where the direction of the inequality was not properly switched.
### Correction:
- Correct final inequality: [tex]\( x < -5 \)[/tex]
Final Correct Inequality:
[tex]\[
x < -5
\][/tex]
