Let's break down the problem step by step to find the correct equation and the constraints on [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
1. Understanding the Problem:
- Vlad spent 20 minutes on his history homework.
- He then solved [tex]\( x \)[/tex] math problems, each taking 2 minutes to complete.
- We need to form an equation to represent the total time [tex]\( y \)[/tex] Vlad spent on his homework.
2. Setting Up the Equation:
- The time spent on history homework is a constant 20 minutes.
- The time spent on math homework is dependent on the number of problems [tex]\( x \)[/tex] that Vlad solved. Each problem takes 2 minutes, so the total time spent on math problems is [tex]\( 2x \)[/tex] minutes.
- The total time [tex]\( y \)[/tex] spent on homework can be expressed as:
[tex]\[
y = 2x + 20
\][/tex]
3. Constraints on [tex]\( x \)[/tex]:
- Since [tex]\( x \)[/tex] represents the number of math problems solved, [tex]\( x \)[/tex] can be any non-negative real number (since it could be a fractional number of problems, although unusual in this context).
- Therefore, the constraint on [tex]\( x \)[/tex] is:
[tex]\[
x \text{ is any real number greater than or equal to 0}
\][/tex]
4. Constraints on [tex]\( y \)[/tex]:
- Since [tex]\( y \)[/tex] is the total time spent on homework, the minimum time [tex]\( y \)[/tex] can be is when Vlad spends 20 minutes on his history homework and solves 0 math problems ([tex]\( x = 0 \)[/tex]):
[tex]\[
y_{\text{min}} = 2(0) + 20 = 20
\][/tex]
- Therefore, the constraint on [tex]\( y \)[/tex] is:
[tex]\[
y \text{ is any real number greater than or equal to 20}
\][/tex]
Given the correct equation and constraints, the correct choice is:
[tex]\[
\boxed{y = 2x + 20 ; x \text{ is any real number greater than or equal to 0, and } y \text{ is any real number greater than or equal to 20.}}
\][/tex]
