Question

lesson 19 | session 4
3 workers at a wildlife center plan a new habitat for meerkats. the habitat must include at least 6.5 m² of space for each meerkat. the area of the habitat is 78 m². how many meerkats can the habitat hold? write, solve, and graph an inequality to find the possible numbers of meerkats. show your work.
solution
4 a summer camp rewards campers and counselors with badges. the camp orders 200 badges. they plan to give 25 badges to counselors. they ordered at least 3 badges for each camper.
a. how many campers could be at the camp? show your work.
solution
b. are all of the values on the graph of c ≤ 58 possible solutions? explain.

Explanation:

Problem 3 (Meerkat Habitat)

Step 1: Define Variable and Inequality

Let \( m \) be the number of meerkats. Each meerkat needs at least \( 6.5 \, \text{m}^2 \), and the total area is \( 78 \, \text{m}^2 \). So the inequality is \( 6.5m \leq 78 \) (since the total space used by meerkats can't exceed the habitat area).

Step 2: Solve the Inequality

Divide both sides by \( 6.5 \): \( m \leq \frac{78}{6.5} \)
Calculate \( \frac{78}{6.5} = 12 \). So \( m \leq 12 \), and since \( m \) is the number of meerkats, \( m \) is a non - negative integer (\( m\geq0 \) and \( m \) is an integer).

Step 3: Graph the Inequality

On a number line, we draw a closed circle at \( 12 \) (because \( m = 12 \) is included) and shade to the left (towards \( 0 \)) to represent all integers from \( 0 \) to \( 12 \).

Step 1: Define Variable and Inequality

Let \( c \) be the number of campers. The camp gives 25 badges to counselors and at least 3 badges to each camper, and they ordered 200 badges. So the inequality is \( 25 + 3c\leq200 \) (the total number of badges given to counselors and campers can't exceed the number of badges ordered).

Step 2: Solve the Inequality

Subtract 25 from both sides: \( 3c\leq200 - 25=175 \)
Divide both sides by 3: \( c\leq\frac{175}{3}\approx58.33 \)
Since \( c \) is the number of campers, \( c \) is a non - negative integer (\( c\geq0 \) and \( c \) is an integer). So \( c\leq58 \) (because \( c \) must be an integer).

No, not all values on the graph of \( c\leq58 \) (as a real - number line graph) are possible solutions. The number of campers \( c \) must be a non - negative integer. For example, values like \( 58.2 \) or \( - 1 \) are on the graph of the inequality \( c\leq58 \) (when considered as a real - valued inequality), but \( c=-1 \) is not a valid number of campers (since the number of campers can't be negative), and \( c = 58.2 \) is not an integer (you can't have a fraction of a camper). Only the non - negative integers from 0 to 58 are possible solutions.

Answer:

The habitat can hold at most 12 meerkats. The inequality is \( 6.5m\leq78 \), the solution is \( m\leq12 \) (where \( m \) is a non - negative integer), and the graph has a closed circle at 12 and shading to the left.

Problem 4a (Summer Camp Badges)