To find an inequality for the number of weeks Lily will need to save, we start by noting her initial amount from her grandfather, which is $40, and her weekly earnings from tutoring, which is $65.
Let's denote \( x \) as the number of weeks she saves. The total amount Lily will have after \( x \) weeks is given by:
\[ \text{Total savings} = 40 + 65x \]
To find out when she can buy the cellphone, we set up the inequality:
\[ 40 + 65x \geq 300 \]
Next, we need to isolate \( x \).
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Subtract 40 from both sides: \[ 65x \geq 300 - 40 \] \[ 65x \geq 260 \]
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Divide both sides by 65: \[ x \geq \frac{260}{65} \] \[ x \geq 4 \]
So, the inequality that represents the number of weeks Lily will need to save is:
\[ x \geq 4 \]
This means that Lily must save for at least 4 weeks to have at least $300.