To determine how many extra hours \( x \) Nathan needs to work to earn at least $312.00, we can set up the equation based on Nathan's pay structure.
- Nathan's fixed pay for a normal 30 hours is $250.25.
- For every extra hour he works, he earns an additional $9.50.
His total pay can be calculated with the following formula:
\[ \text{Total Pay} = 250.25 + 9.50 \times x \]
To find out how many extra hours Nathan needs to work to earn at least $312.00, we set up the inequality:
\[ 250.25 + 9.50x \geq 312.00 \]
Now, we will isolate \( x \):
- Subtract $250.25 from both sides:
\[ 9.50x \geq 312.00 - 250.25 \]
- Calculate the right side:
\[ 9.50x \geq 61.75 \]
- Now, divide both sides by 9.50:
\[ x \geq \frac{61.75}{9.50} \]
- Performing the division gives:
\[ x \geq 6.5 \]
This means Nathan needs to work at least 6.5 extra hours in order to earn at least $312.00.
To represent this graphically, you would plot the line \( x = 6.5 \) on a number line and shade to the right (indicating that he needs to work 6.5 hours or more).
The graph of the inequality will show that any value of \( x \) greater than or equal to 6.5 is acceptable.