Answer :
To solve the given system of inequalities:
[tex]\[ \begin{aligned} y & \leq \frac{4}{3} x - 2 \\ x & \leq 3 \end{aligned} \][/tex]
we will analyze each inequality step-by-step.
### Step 1: Understand the Inequality [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex]
First, consider the linear inequality [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex]. This can be interpreted as a region below the line defined by the equation [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
#### Finding the line's intercepts:
- Y-intercept: Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = \frac{4}{3}(0) - 2 = -2 \][/tex]
So, the y-intercept is [tex]\( (0, -2) \)[/tex].
- X-intercept: Set [tex]\( y = 0 \)[/tex] in the equation:
[tex]\[ 0 = \frac{4}{3}x - 2 \][/tex]
Solving for [tex]\( x \)[/tex], we have:
[tex]\[ 2 = \frac{4}{3}x \implies x = \frac{3}{2} \times 2 = \frac{3 \times 2}{4} = 1.5 \][/tex]
So, the x-intercept is [tex]\( (1.5, 0) \)[/tex].
Hence, the line defined by [tex]\( y = \frac{4}{3} x - 2 \)[/tex] passes through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (1.5, 0) \)[/tex].
### Step 2: Understand the Inequality [tex]\( x \leq 3 \)[/tex]
This inequality represents a vertical line at [tex]\( x = 3 \)[/tex]. The region of interest includes all points to the left of this line (i.e., where [tex]\( x \)[/tex] is less than or equal to 3).
### Step 3: Combine the Inequalities
To find the region that satisfies both inequalities, we need to consider the intersection of these regions:
- The area below the line [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex].
- The area to the left of the line [tex]\( x \leq 3 \)[/tex].
### Step 4: Identify Key Points and Graph the Region
1. Points on the Line [tex]\( x \leq 3 \)[/tex]:
- Calculate [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{4}{3}(3) - 2 = 4 - 2 = 2 \][/tex]
So, the point [tex]\( (3, 2) \)[/tex] lies on the line [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
2. Assess the Region Below the Line:
- Consider points such as [tex]\( (1, \frac{4}{3}(1) - 2 = \frac{4}{3} - 2 = -\frac{2}{3}) \)[/tex].
- Another example could be [tex]\( (0, -2) \)[/tex], as calculated earlier.
3. Combine the Regions:
- Points where [tex]\( x \leq 3 \)[/tex] and also fall below (or on) the line [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
### Region of Interest:
To summarize, the solution region is the area where [tex]\( x \)[/tex] ranges from [tex]\( -\infty \)[/tex] to 3, and [tex]\( y \)[/tex] values fall below or on the line [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
#### Key Vertices for the Region of Interest:
- [tex]\( (0, -2) \)[/tex]: Derived from the y-intercept.
- [tex]\( (3, 2) \)[/tex]: Derived from setting [tex]\( x = 3 \)[/tex] in the line equation.
- [tex]\( (1.5, 0) \)[/tex]: The x-intercept of the line.
In graphical terms, the solution can be described as the area below the line starting from the y-intercept at [tex]\( (0, -2) \)[/tex], through the x-intercept at [tex]\( (1.5, 0) \)[/tex], up to the point [tex]\( (3, 2) \)[/tex], including the segment of the vertical line [tex]\( x = 3 \)[/tex].
### Final Representation:
- The solution region includes all points [tex]\( (x, y) \)[/tex] such that [tex]\( x \leq 3 \)[/tex] and [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex].
[tex]\[ \begin{aligned} y & \leq \frac{4}{3} x - 2 \\ x & \leq 3 \end{aligned} \][/tex]
we will analyze each inequality step-by-step.
### Step 1: Understand the Inequality [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex]
First, consider the linear inequality [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex]. This can be interpreted as a region below the line defined by the equation [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
#### Finding the line's intercepts:
- Y-intercept: Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = \frac{4}{3}(0) - 2 = -2 \][/tex]
So, the y-intercept is [tex]\( (0, -2) \)[/tex].
- X-intercept: Set [tex]\( y = 0 \)[/tex] in the equation:
[tex]\[ 0 = \frac{4}{3}x - 2 \][/tex]
Solving for [tex]\( x \)[/tex], we have:
[tex]\[ 2 = \frac{4}{3}x \implies x = \frac{3}{2} \times 2 = \frac{3 \times 2}{4} = 1.5 \][/tex]
So, the x-intercept is [tex]\( (1.5, 0) \)[/tex].
Hence, the line defined by [tex]\( y = \frac{4}{3} x - 2 \)[/tex] passes through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (1.5, 0) \)[/tex].
### Step 2: Understand the Inequality [tex]\( x \leq 3 \)[/tex]
This inequality represents a vertical line at [tex]\( x = 3 \)[/tex]. The region of interest includes all points to the left of this line (i.e., where [tex]\( x \)[/tex] is less than or equal to 3).
### Step 3: Combine the Inequalities
To find the region that satisfies both inequalities, we need to consider the intersection of these regions:
- The area below the line [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex].
- The area to the left of the line [tex]\( x \leq 3 \)[/tex].
### Step 4: Identify Key Points and Graph the Region
1. Points on the Line [tex]\( x \leq 3 \)[/tex]:
- Calculate [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{4}{3}(3) - 2 = 4 - 2 = 2 \][/tex]
So, the point [tex]\( (3, 2) \)[/tex] lies on the line [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
2. Assess the Region Below the Line:
- Consider points such as [tex]\( (1, \frac{4}{3}(1) - 2 = \frac{4}{3} - 2 = -\frac{2}{3}) \)[/tex].
- Another example could be [tex]\( (0, -2) \)[/tex], as calculated earlier.
3. Combine the Regions:
- Points where [tex]\( x \leq 3 \)[/tex] and also fall below (or on) the line [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
### Region of Interest:
To summarize, the solution region is the area where [tex]\( x \)[/tex] ranges from [tex]\( -\infty \)[/tex] to 3, and [tex]\( y \)[/tex] values fall below or on the line [tex]\( y = \frac{4}{3} x - 2 \)[/tex].
#### Key Vertices for the Region of Interest:
- [tex]\( (0, -2) \)[/tex]: Derived from the y-intercept.
- [tex]\( (3, 2) \)[/tex]: Derived from setting [tex]\( x = 3 \)[/tex] in the line equation.
- [tex]\( (1.5, 0) \)[/tex]: The x-intercept of the line.
In graphical terms, the solution can be described as the area below the line starting from the y-intercept at [tex]\( (0, -2) \)[/tex], through the x-intercept at [tex]\( (1.5, 0) \)[/tex], up to the point [tex]\( (3, 2) \)[/tex], including the segment of the vertical line [tex]\( x = 3 \)[/tex].
### Final Representation:
- The solution region includes all points [tex]\( (x, y) \)[/tex] such that [tex]\( x \leq 3 \)[/tex] and [tex]\( y \leq \frac{4}{3} x - 2 \)[/tex].
