Answered

Discover a world of knowledge and community-driven answers at IDNLearn.com today. Discover comprehensive answers to your questions from our community of knowledgeable experts.

The following inequalities represent a system:

[tex]\( \begin{cases}
y \geq -5x + 2 \\
y \ \textgreater \ 3x - 1.5
\end{cases} \)[/tex]

Which of the following graphs represents the system?

Sagot :

GinnyAnsweridn
To solve this system of inequalities, we need to analyze and graph the two lines represented by:

1. [tex]\( y = -5x + 2 \)[/tex]
2. [tex]\( y = 3x - 1.5 \)[/tex]

### Step-by-Step Solution

1. Find the intersection point of the two lines:
To find where the two lines intersect, we set the equations equal to each other:
[tex]\[ -5x + 2 = 3x - 1.5 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ -5x - 3x + 2 = -1.5 \\ -8x + 2 = -1.5 \\ -8x = -1.5 - 2 \\ -8x = -3.5 \\ x = \frac{-3.5}{-8} \\ x = 0.4375 \][/tex]

Now, solve for [tex]\( y \)[/tex] using [tex]\( x = 0.4375 \)[/tex] in either of the original equations. Using [tex]\( y = -5x + 2 \)[/tex]:
[tex]\[ y = -5(0.4375) + 2 \\ y = -2.1875 + 2 \\ y = -0.1875 \][/tex]

The intersection point is [tex]\( (0.4375, -0.1875) \)[/tex].

2. Graph the lines:
- The line [tex]\( y = -5x + 2 \)[/tex]: This line has a slope of -5 and a y-intercept at [tex]\( (0, 2) \)[/tex].
- The line [tex]\( y = 3x - 1.5 \)[/tex]: This line has a slope of 3 and a y-intercept at [tex]\( (0, -1.5) \)[/tex].

3. Shading the regions:
- For the inequality [tex]\( y \geq -5x + 2 \)[/tex], shade the region above or on the line [tex]\( y = -5x + 2 \)[/tex].
- For the inequality [tex]\( y > 3x - 1.5 \)[/tex], shade the region strictly above the line [tex]\( y = 3x - 1.5 \)[/tex] (note that this does not include the line itself).

4. Determine the feasible region:
- The feasible region is the area that satisfies both inequalities simultaneously. This is the region above the line [tex]\( y = -5x + 2 \)[/tex] AND strictly above [tex]\( y = 3x - 1.5 \)[/tex].

### Final Graph
The graph should include:
- The line [tex]\( y = -5x + 2 \)[/tex] in solid, indicating the boundary for the [tex]\(\geq\)[/tex] inequality, with shading above the line including the line itself.
- The line [tex]\( y = 3x - 1.5 \)[/tex] in dashed, indicating the boundary for the [tex]\(>\)[/tex] inequality, with shading strictly above the line excluding the line itself.
- The intersection point at [tex]\( (0.4375, -0.1875) \)[/tex].

Look for the graph that illustrates the above description.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.