Start with
![12h+240>360](https://img.qammunity.org/2020/formulas/mathematics/middle-school/b1jab0s072lavrmbbjw5pcnht7d22xcuwc.png)
Subtract 240 from both sides:
![12h>120](https://img.qammunity.org/2020/formulas/mathematics/middle-school/k14b92y733e72eczppiunhox3uazeum8qz.png)
Divide both sides by 12:
![h>10](https://img.qammunity.org/2020/formulas/mathematics/middle-school/vz6ze61r5bnxq80mjhxrclp1wxlu8u1ufr.png)
So, she needs to work at least 10 hours. Note that this is the minimum number of hours Ellie needs to work this week to earn at least $360 by the end of the month. This problem has no maximum: the more she works, the more she gains, so there's likely a typo in the question.