To determine which system of inequalities includes a line as a solution, let's analyze each system of inequalities step-by-step.
1. System 1:
[tex]\( 2x + 4y \geq 3 \)[/tex]
This inequality describes a half-plane including the boundary line described by [tex]\( 2x + 4y = 3 \)[/tex].
2. System 2:
[tex]\( 2x + 4y \leq 3 \)[/tex]
This inequality describes another half-plane including the boundary line described by [tex]\( 2x + 4y = 3 \)[/tex].
3. System 3:
[tex]\[
\begin{aligned}
2x + 4y & \geq 3 \\
2x + 4y & > 3
\end{aligned}
\][/tex]
The first inequality [tex]\( 2x + 4y \geq 3 \)[/tex] includes the boundary line [tex]\( 2x + 4y = 3 \)[/tex], but the second inequality [tex]\( 2x + 4y > 3 \)[/tex] excludes this line. Therefore, this system does not have a line as a solution because the overlapping region strictly contains values where [tex]\( 2x + 4y > 3 \)[/tex], excluding the line [tex]\( 2x + 4y = 3 \)[/tex].
4. System 4:
[tex]\( 2x + 4y > 3 \)[/tex]
This inequality describes a half-plane excluding the boundary line [tex]\( 2x + 4y = 3 \)[/tex].
5. System 5:
[tex]\( 2x + 4y < 3 \)[/tex]
This inequality describes another half-plane excluding the boundary line [tex]\( 2x + 4y = 3 \)[/tex].
Since we are interested in the system of inequalities that includes a line as a solution, we need to find the inequality or a set of inequalities that represent a half-plane including its boundary line.
Upon reviewing each system:
- System 1: [tex]\( 2x + 4y \geq 3 \)[/tex] includes the boundary line [tex]\( 2x + 4y = 3 \)[/tex].
- System 2: [tex]\( 2x + 4y \leq 3 \)[/tex] also includes the boundary line [tex]\( 2x + 4y = 3 \)[/tex].
Thus, when assessing which of these systems most clearly includes the line as part of its solution, we find the first system meets this criterion.
Therefore, the system of inequalities that has a line as a solution is:
[tex]\[ \boxed{1} \][/tex]
