To determine the graphical solution for the given system of inequalities:
[tex]$
\begin{array}{r}
4x - 2y \geq -4 \\
2x + y > -3
\end{array}
$[/tex]
we need to follow these steps:
### Simplify and Rewrite Each Inequality
1. First Inequality: [tex]\(4x - 2y \geq -4\)[/tex]
Simplify this inequality by dividing all terms by 2:
[tex]$2x - y \geq -2$[/tex]
Rearrange to solve for [tex]\(y\)[/tex]:
[tex]$-y \geq -2x - 2$[/tex]
Multiply through by -1 (remember to reverse the inequality sign):
[tex]$y \leq 2x + 2$[/tex]
2. Second Inequality: [tex]\(2x + y > -3\)[/tex]
Rearrange to solve for [tex]\(y\)[/tex]:
[tex]$y > -2x - 3$[/tex]
### Plot Each Inequality on the Graph
- First Inequality: [tex]\(y \leq 2x + 2\)[/tex]
- Draw the line [tex]\(y = 2x + 2\)[/tex] (a line with a slope of 2 and y-intercept at (0, 2)).
- Because the inequality is [tex]\(y \leq 2x + 2\)[/tex], shade the region below this line.
- Second Inequality: [tex]\(y > -2x - 3\)[/tex]
- Draw the line [tex]\(y = -2x - 3\)[/tex] (a line with a slope of -2 and y-intercept at (0, -3)).
- Because the inequality is [tex]\(y > -2x - 3\)[/tex], shade the region above this line.
### Determine the Solution Region
The solution to the system of inequalities is the region where the shaded areas of the two inequalities overlap. This is the region where both conditions [tex]\(y \leq 2x + 2\)[/tex] and [tex]\(y > -2x - 3\)[/tex] are simultaneously satisfied.
### Match with Answer Options
To identify the correct graph:
1. Look at the given answer options.
2. Identify which graph correctly shows:
- The line [tex]\(y = 2x + 2\)[/tex] with shading below it.
- The line [tex]\(y = -2x - 3\)[/tex] with shading above it.
- The overlapping region where both conditions are true.
The correct graph will depict these regions correctly, highlighting the area of intersection as the solution to the system of inequalities.
