To determine which inequality Hessa solved, we need to analyze the closed point at 5 and the leftward arrow indicating that the solution includes numbers less than or equal to 5.
- A closed point at 5 means that the solution includes 5 itself.
- The leftward arrow indicates that the values extend to negative infinity, meaning the solution should be of the form \( x \leq 5 \).
Now, let's check each of the given inequalities to see which one results in \( x \leq 5 \):
-
\( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \):
- Multiply all terms by 5: \( x - 3 \leq 2 \)
- Adding 3 to both sides: \( x \leq 5 \)
This matches the graphical representation.
-
\( \frac{3}{5}x + \frac{2}{5} \geq 3 \frac{2}{5} \):
- Rewrite \( 3 \frac{2}{5} \) as \( \frac{17}{5} \).
- Multiply all terms by 5: \( 3x + 2 \geq 17 \)
- Subtracting 2 gives: \( 3x \geq 15 \)
- Dividing by 3: \( x \geq 5 \)
This does not match the graphical representation.
-
\( \frac{3}{7}x + \frac{1}{7} \leq 1 \frac{6}{7} \):
- Rewrite \( 1 \frac{6}{7} \) as \( \frac{13}{7} \).
- Multiply all terms by 7: \( 3x + 1 \leq 13 \)
- Subtracting 1 gives: \( 3x \leq 12 \)
- Dividing by 3: \( x \leq 4 \)
This does not match the graphical representation.
-
\( \frac{2}{5}x - \frac{4}{5} \geq 1 \frac{1}{5} \):
- Rewrite \( 1 \frac{1}{5} \) as \( \frac{6}{5} \).
- Multiply all terms by 5: \( 2x - 4 \geq 6 \)
- Adding 4 to both sides: \( 2x \geq 10 \)
- Dividing by 2: \( x \geq 5 \)
This does not match the graphical representation.
Based on the analysis, the inequality Hessa solved that matches the closed point at 5 and extends leftward is:
\( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \).