10. Open Response Robin was given a $$ \$ 40 $$ monthly allowance. She wants to go to the movies as many times as possible and have at least $$ \$ 12.50 $$ left at the end of the month to go to a concert. A movie ticket costs $$ \$ 5 $$. Write and solve an inequality to determine how many times Robin can go to the movies this month. Then interpret the solution.

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Let $$ x $$ represent the number of times Robin goes to the movies. She spends $$ \$5 $$ per movie so her total spending on movies is $$ 5x $$. Since she wants at least $$ \$12.50 $$ left from her $$ \$40 $$ allowance, we set up the inequality:

$$
40 - 5x \geq 12.50
$$

Subtract $$ 40 $$ from both sides:

$$
-5x \geq 12.50 - 40
$$

$$
-5x \geq -27.50
$$

Divide both sides by $$-5$$ and reverse the inequality sign (since dividing by a negative number reverses the inequality):

$$
x \leq \frac{-27.50}{-5}
$$

$$
x \leq 5.5
$$

Since $$ x $$ must be an integer (she can't go to the movies a fractional number of times), the greatest number of whole movies Robin can attend is $$ 5 $$.

Interpretation: Robin can go to the movies at most $$ 5 $$ times while still having at least $$ \$12.50 $$ left for the concert.
Robin can go to the movies up to 5 times this month and still have at least \$12.50 left for the concert.

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