Sure! Let's solve and understand the given system of inequalities step-by-step to identify the correct graph.
The system of inequalities is:
[tex]\[
\left\{\begin{array}{l}
y < -2x + 4 \\
y < x + 3 \\
x < 3
\end{array}\right.
\][/tex]
We will analyze each inequality one at a time and sketch their corresponding boundary lines before determining the solution region.
1. First Inequality: [tex]\( y < -2x + 4 \)[/tex]
- The boundary line is [tex]\( y = -2x + 4 \)[/tex].
- This is a line with a slope of -2 and a y-intercept at 4.
- Since it is a '<' inequality, the region below this line is shaded.
2. Second Inequality: [tex]\( y < x + 3 \)[/tex]
- The boundary line is [tex]\( y = x + 3 \)[/tex].
- This is a line with a slope of 1 and a y-intercept at 3.
- Since it is a '<' inequality, the region below this line is shaded.
3. Third Inequality: [tex]\( x < 3 \)[/tex]
- The boundary line is [tex]\( x = 3 \)[/tex].
- This is a vertical line passing through [tex]\( x = 3 \)[/tex].
- Since it is a '<' inequality, the region to the left of this line is shaded.
To find the solution to the system of inequalities, we need to find the region where all these conditions overlap. To do this:
1. Sketch the line [tex]\( y = -2x + 4 \)[/tex] and shade the area below it.
2. Sketch the line [tex]\( y = x + 3 \)[/tex] and shade the area below it.
3. Draw the vertical line [tex]\( x = 3 \)[/tex] and shade the area to its left.
The overlapping shaded region of these inequalities is the solution to the system.
The solution region shows the common area that satisfies:
[tex]\[
\left\{\begin{array}{l}
y < -2x + 4 \\
y < x + 3 \\
x < 3
\end{array}\right.
\][/tex]
Given the numerical result (0, 0, 0, 0), the points on the boundary lines and respective shaded regions must all satisfy these inequalities collectively.
Hence, the graph that correctly represents the solution to this system of inequalities is identified by the overlapping shaded region as described.
