Answer :
- Define variables: $x$ = infected, $y$ = healthy.
- Total population: $x + y$.
- Inequality: $x
ge 0.15(x + y)$.
- Simplify: $17x
ge 3y$.
- Final answer: $\boxed{17 x
ge 3 y}$.
### Explanation
1. Setting up the inequality
Let $x$ be the number of infected cervids and $y$ be the number of healthy cervids. The total number of cervids is $x+y$. The statement "at least $15\%$ of the cervids in this population are infected" can be written as $x \geq 0.15(x+y)$.
2. Simplifying the inequality
Now we simplify the inequality:
$x \geq 0.15(x+y)$
$x \geq 0.15x + 0.15y$
$x - 0.15x \geq 0.15y$
$0.85x \geq 0.15y$
3. Removing decimals
Multiply both sides by 100 to remove decimals:
$85x \geq 15y$
4. Simplifying coefficients
Divide both sides by 5 to simplify the coefficients:
$17x \geq 3y$
5. Final Answer
The inequality that describes the statement is $17x \geq 3y$. This corresponds to option (a).
### Examples
Understanding percentages and inequalities is crucial in epidemiology to model the spread of diseases. For example, public health officials might use similar inequalities to determine the minimum vaccination rate needed to achieve herd immunity in a population. By setting up and solving inequalities, they can estimate the number of individuals who need to be vaccinated to protect the entire community from a disease outbreak. This helps in planning vaccination campaigns and allocating resources effectively.
- Total population: $x + y$.
- Inequality: $x
ge 0.15(x + y)$.
- Simplify: $17x
ge 3y$.
- Final answer: $\boxed{17 x
ge 3 y}$.
### Explanation
1. Setting up the inequality
Let $x$ be the number of infected cervids and $y$ be the number of healthy cervids. The total number of cervids is $x+y$. The statement "at least $15\%$ of the cervids in this population are infected" can be written as $x \geq 0.15(x+y)$.
2. Simplifying the inequality
Now we simplify the inequality:
$x \geq 0.15(x+y)$
$x \geq 0.15x + 0.15y$
$x - 0.15x \geq 0.15y$
$0.85x \geq 0.15y$
3. Removing decimals
Multiply both sides by 100 to remove decimals:
$85x \geq 15y$
4. Simplifying coefficients
Divide both sides by 5 to simplify the coefficients:
$17x \geq 3y$
5. Final Answer
The inequality that describes the statement is $17x \geq 3y$. This corresponds to option (a).
### Examples
Understanding percentages and inequalities is crucial in epidemiology to model the spread of diseases. For example, public health officials might use similar inequalities to determine the minimum vaccination rate needed to achieve herd immunity in a population. By setting up and solving inequalities, they can estimate the number of individuals who need to be vaccinated to protect the entire community from a disease outbreak. This helps in planning vaccination campaigns and allocating resources effectively.