To graph the solution to the linear inequality [tex]\( 2x + y > 4 \)[/tex], follow these steps:
### 1. Convert the Inequality to an Equation:
First, let's start with the boundary line [tex]\( 2x + y = 4 \)[/tex]. Graphing this boundary line will help us identify the region that satisfies the inequality.
### 2. Plot the Boundary Line:
The boundary line [tex]\( 2x + y = 4 \)[/tex] can be graphed by finding its intercepts and then plotting those points.
Finding the intercepts:
- x-intercept: When [tex]\( y = 0 \)[/tex]:
[tex]\[
2x + 0 = 4 \implies x = 2
\][/tex]
So, the x-intercept is [tex]\( (2, 0) \)[/tex].
- y-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[
2(0) + y = 4 \implies y = 4
\][/tex]
So, the y-intercept is [tex]\( (0, 4) \)[/tex].
Plot these points on a coordinate plane: Plot the points [tex]\( (2, 0) \)[/tex] and [tex]\( (0, 4) \)[/tex].
Draw the boundary line: Connect these points with a straight line. Since the inequality is [tex]\( 2x + y > 4 \)[/tex] (strict inequality), this boundary line will be dashed to indicate that points on the line are not included in the solution.
### 3. Determine Which Side of the Line to Shade:
To find out which side of the line [tex]\( 2x + y = 4 \)[/tex] satisfies the inequality [tex]\( 2x + y > 4 \)[/tex], we can use a test point that is not on the line. The origin [tex]\( (0, 0) \)[/tex] is a convenient test point.
- Test point [tex]\( (0, 0) \)[/tex]:
[tex]\[
2(0) + 0 > 4 \implies 0 > 4
\][/tex]
This statement is false, so the region that does not include the origin will be the solution to our inequality.
### 4. Shade the Solution Region:
Shade the region above the dashed line [tex]\( 2x + y = 4 \)[/tex] because this is the region where [tex]\( 2x + y > 4 \)[/tex].
### 5. Final Graph:
The final graph will have:
- A dashed line passing through [tex]\( (2, 0) \)[/tex] and [tex]\( (0, 4) \)[/tex],
- And the region above this line shaded, representing the solution set to the inequality [tex]\( 2x + y > 4 \)[/tex].
Here is how the graph looks:
[tex]\[
\begin{array}{c}
\begin{tikzpicture}[scale=0.6]
% Draw axes
\draw[thin, gray!30] (-10,-10) grid (10,10);
\draw[<->] (-10,0) -- (10,0) node[right] {$x$};
\draw[<->] (0,-10) -- (0,10) node[above] {$y$};
% Draw the boundary line
\draw[dashed, thick, red] (-5,14) -- (6,-8);
% Add intercepts
\fill (2,0) circle (2pt) node[below right] {$(2,0)$};
\fill (0,4) circle (2pt) node[above left] {$(0,4)$};
% Shade the region
\fill[red!20] (-5,14) -- (6,-8) -- (10,10) -- (-10,10) -- cycle;
\end{tikzpicture}
\end{array}
\][/tex]
This graph visually represents all the points [tex]\((x, y)\)[/tex] that satisfy the inequality [tex]\( 2x + y > 4 \)[/tex].
