Certainly! Let's analyze and solve the given inequality step by step.
### Step 1: Understanding the Inequality
The given inequality is:
[tex]\[ y \geq -3x + 4 \][/tex]
This inequality represents all the points [tex]\((x, y)\)[/tex] in the coordinate plane that lie on or above the line described by the equation:
[tex]\[ y = -3x + 4 \][/tex]
### Step 2: Graphing the Line
To understand the inequality fully, we should first consider the line itself. The line [tex]\( y = -3x + 4 \)[/tex] can be graphed by identifying two key points:
1. Y-Intercept: The line intersects the y-axis at [tex]\( y = 4 \)[/tex]. So, one point on the line is [tex]\((0, 4)\)[/tex].
2. X-Intercept: To find where the line crosses the x-axis, we set [tex]\( y = 0 \)[/tex]:
[tex]\[
0 = -3x + 4 \implies 3x = 4 \implies x = \frac{4}{3}
\][/tex]
Thus, the x-intercept is [tex]\((\frac{4}{3}, 0)\)[/tex].
### Step 3: Plotting the Line
On the coordinate plane, plot the points [tex]\((0, 4)\)[/tex] and [tex]\((\frac{4}{3}, 0)\)[/tex]. Draw a straight line through these points. This line is solid because the inequality involves a "greater than or equal to" (≥), indicating that points on the line are included in the solution set of the inequality.
### Step 4: Shading the Region
According to the inequality [tex]\( y \geq -3x + 4 \)[/tex], we want all points [tex]\((x, y)\)[/tex] where the y-coordinate is greater than or equal to the value of [tex]\(-3x + 4\)[/tex]. This corresponds to the region above the line (including the line itself).
To determine if your shading is correct, you can test a point not on the line. For example, take the point [tex]\((0,5)\)[/tex]:
Plug into the inequality:
[tex]\[ 5 \geq -3(0) + 4 \implies 5 \geq 4 \][/tex]
This is true, so the point [tex]\((0, 5)\)[/tex] is part of the solution set, confirming that the region above the line [tex]\( y = -3x + 4 \)[/tex] should be shaded.
### Step 5: Conclusion
In conclusion, the solution set to the inequality [tex]\( y \geq -3x + 4 \)[/tex] includes the line [tex]\( y = -3x + 4 \)[/tex] itself and all points above this line.
The final graphical representation would be a solid line passing through the points [tex]\((0, 4)\)[/tex] and [tex]\((\frac{4}{3}, 0)\)[/tex], with the entire area above this line shaded to indicate that it is included in the solution set.
