To determine the minimum number of hours \( h \) that Hanna must work to earn at least $300 during one week, we first need to set up the inequality based on her earnings.
Hanna's total earnings can be represented as:
\[ \text{Total Earnings} = \text{Base Salary} + \text{Hourly Wage} \times \text{Hours Worked} \] \[ \text{Total Earnings} = 75 + 9h \]
We want to find the minimum \( h \) such that her total earnings are at least $300. Therefore, we can write the inequality:
\[ 75 + 9h \geq 300 \]
Now, we will solve the inequality for \( h \):
- Subtract 75 from both sides:
\[ 9h \geq 300 - 75 \] \[ 9h \geq 225 \]
- Divide both sides by 9:
\[ h \geq \frac{225}{9} \] \[ h \geq 25 \]
Therefore, Hanna must work at least 25 hours to earn $300 or more in a week.
Graphing the Solution:
To graph the solution on a number line:
- Draw a horizontal line representing the number line.
- Mark a point corresponding to the value \( h = 25 \).
- Since \( h \) can be equal to 25 or greater, we will use a closed circle at 25 to indicate that this value is included in the solution.
- Shade the line to the right of 25 to indicate all numbers greater than 25.
The graph should look like this:
<---|----|-----|-----|----|----|----|----|----|----|----|----|----|----|---->
20 21 22 23 24 25 26 27 28 29 30 31 32
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This indicates that any hour count of 25 or more will satisfy the inequality.
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