To determine which inequality Hessa solved based on the graph of the number line, we need to interpret the graphical representation:
- There is a closed point at 5, which means that the solution includes the number 5.
- The leftward arrow indicates that the solution also includes all numbers less than or equal to 5.
This suggests that the inequality represents values that are less than or equal to 5.
Now let's check each of the provided inequalities to see which one leads to a solution that includes all numbers less than or equal to 5:
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\(15x - 35 \leq 25\):
- Adding 35 to both sides gives \(15x \leq 60\).
- Dividing both sides by 15 gives \(x \leq 4\). This does not include 5.
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\(\frac{1}{5}x - \frac{3}{5} < \frac{2}{5}\):
- Adding \(\frac{3}{5}\) to both sides gives \(\frac{1}{5}x < 1\).
- Multiplying by 5 gives \(x < 5\). This does not include 5.
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\(25x - 45 \geq 115\):
- Adding 45 to both sides gives \(25x \geq 160\).
- Dividing both sides by 25 gives \(x \geq 6.4\). This does not include 5.
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\(\frac{2}{5}x - \frac{4}{5} \geq 1 \frac{1}{5}\):
- Converting \(1 \frac{1}{5}\) to an improper fraction gives \(1 \frac{1}{5} = \frac{6}{5}\).
- So we have \(\frac{2}{5}x - \frac{4}{5} \geq \frac{6}{5}\).
- Adding \(\frac{4}{5}\) to both sides gives \(\frac{2}{5}x \geq 2\).
- Multiplying by \(5\) gives \(2x \geq 10\), thus \(x \geq 5\). This does include 5.
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\(37x + 17 \leq 167\):
- Subtracting 17 gives \(37x \leq 150\).
- Dividing by 37 gives \(x \leq \frac{150}{37}\) or approximately \(4.05\). This does not include 5.
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\(35x + 25 \geq 325\):
- Subtracting 25 gives \(35x \geq 300\).
- Dividing by 35 gives \(x \geq \frac{300}{35}\) or approximately \(8.57\). This does not include 5.
Based on this analysis, the only inequality that allows for 5 to be included and has a leftward solution is:
\(\frac{2}{5}x - \frac{4}{5} \geq 1 \frac{1}{5}\).
Therefore, Hessa solved the inequality \( \frac{2}{5}x - \frac{4}{5} \geq 1 \frac{1}{5} \).