Write an inequality representing the allowable weight for a female 5 foot 9 inch mission specialist, given 101 lbs for the first 5 feet, and [tex]$1-5 \text{ lb/in}$[/tex] for every additional inch. Let [tex]$w$[/tex] be the astronaut's weight in pounds.

Fill in the following inequality:

[tex] 101 \leq w \leq 146 [/tex]

Sure! Let's solve the problem step-by-step.

We need to determine the allowable weight range for a female astronaut who is 5 feet 9 inches tall. According to the problem:

1. For the first 5 feet (which is 60 inches), the base weight is 101 pounds.
2. For each additional inch beyond 5 feet, the weight can increase by 1 to 5 pounds.

Since the astronaut is 5 feet 9 inches tall, she is 9 inches taller than 5 feet. We need to account for this additional height:

- Minimum Allowable Additional Weight:
- If the weight increases by 1 pound per inch, then for 9 inches, it's 9 pounds. Thus, the minimum total weight is:
[tex]\[
101 \text{ lbs (base for 5 feet)} + 9 \text{ lbs (minimum additional)} = 110 \text{ lbs}
\][/tex]

- Maximum Allowable Additional Weight:
- If the weight increases by 5 pounds per inch, then for 9 inches, it's 45 pounds. Thus, the maximum total weight is:
[tex]\[
101 \text{ lbs (base for 5 feet)} + 45 \text{ lbs (maximum additional)} = 146 \text{ lbs}
\][/tex]

Therefore, the inequality representing the allowable weight [tex]$ w $[/tex] for the astronaut is:

[tex]\[
110 \leq w \leq 146
\][/tex]

This means the astronaut's weight must be between 110 and 146 pounds.