To solve the inequality [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex], let's break it down step by step.
### Step 1: Understanding the Inequality
Firstly, the inequality given is:
[tex]\[ y \leq -\frac{2}{5} x + 2 \][/tex]
This inequality describes a region on the coordinate plane. The line [tex]\( y = -\frac{2}{5} x + 2 \)[/tex] is the boundary of that region.
### Step 2: Identify the Line Equation
The line equation from the inequality is:
[tex]\[ y = -\frac{2}{5} x + 2 \][/tex]
Here, the slope of the line ([tex]\( m \)[/tex]) is [tex]\( -\frac{2}{5} \)[/tex], and the y-intercept ([tex]\( b \)[/tex]) is [tex]\( 2 \)[/tex].
### Step 3: Graph the Boundary Line
To graph the line [tex]\( y = -\frac{2}{5} x + 2 \)[/tex]:
1. Y-Intercept: Start by plotting the y-intercept at [tex]\( (0, 2) \)[/tex] on the coordinate plane.
2. Slope: Use the slope [tex]\( -\frac{2}{5} \)[/tex] to find another point on the line. From the y-intercept [tex]\( (0, 2) \)[/tex], move down 2 units (because of the -2 in the numerator of the slope) and right 5 units (because of the 5 in the denominator). This gives you another point at [tex]\( (5, 0) \)[/tex].
3. Draw the Line: Draw a straight line through these points. This line represents [tex]\( y = -\frac{2}{5} x + 2 \)[/tex].
### Step 4: Determine the Shaded Region
Since the inequality is [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex], the region of interest is the area where [tex]\( y \)[/tex] is less than or equal to the values on the line.
- This means you will shade the region below the line [tex]\( y = -\frac{2}{5} x + 2 \)[/tex].
### Step 5: Verify with a Test Point
To ensure you've shaded the correct side, you can choose a test point not on the line. A commonly used test point is [tex]\( (0, 0) \)[/tex].
Substitute [tex]\( (0, 0) \)[/tex] into the inequality:
[tex]\[ 0 \leq -\frac{2}{5} (0) + 2 \][/tex]
[tex]\[ 0 \leq 2 \][/tex]
The test point [tex]\( (0, 0) \)[/tex] satisfies the inequality, confirming that the region below the line is correctly shaded.
### Conclusion
The solution to the inequality [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex] is represented graphically by the line [tex]\( y = -\frac{2}{5} x + 2 \)[/tex] and the shaded region below this line.
