# How do you solve this?

## In Gym class, Sue is shooting basketball free throws. She is shooting three sets of twenty shots each. In her first two attempts, she made 13 and 18 free throws. What are the possible scores on her third attempt if she is to have an average between 12 and 16 free throws after all three attempts? Express your answer in the form of a compound inequality using the variable F to represent a successful basketball free throw.

Jan 15, 2017

$5 \le F \le 17$

#### Explanation:

If ${s}_{1} , {s}_{2} , {s}_{3}$ her scores (amount of free throws made) on each set respectively, her average score is:

$a = \frac{{s}_{1} + {s}_{2} + {s}_{3}}{3}$.

The average needs to be between $12$ and $16$, so $12 \le a \le 16$

Multiplying by $3$ to get rid of the denominator:

$12 \le \frac{{s}_{1} + {s}_{2} + {s}_{3}}{3} \le 16 \implies 36 \le {s}_{1} + {s}_{2} + {s}_{3} \le 48$.

We know that ${s}_{1} = 13$ and ${s}_{2} = 18$, so

$36 \le 31 + {s}_{3} \le 48$.

${s}_{3}$ is actually what the problem wants us to use as $F$, so let's do that.

Subtracting $31$ from the inequality gives the solution:

$5 \le F \le 17$.

That means the set of all possible scores is

$S = \left\{5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17\right\}$