Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

[tex]\[
\begin{array}{l}
y \ \textless \ -\frac{3}{2} x - 7 \\
y \ \textless \ \frac{1}{2} x + 1
\end{array}
\][/tex]

Answer:

Point: [tex]\square[/tex]

Sagot :

To solve the system of inequalities graphically, we need to consider the following inequalities:
1. [tex]\( y < -\frac{3}{2} x - 7 \)[/tex]
2. [tex]\( y < \frac{1}{2} x + 1 \)[/tex]

Here's the step-by-step solution:

1. Graph the boundary lines:
- For the inequality [tex]\( y < -\frac{3}{2} x - 7 \)[/tex], we first draw the line [tex]\( y = -\frac{3}{2} x - 7 \)[/tex]. This line has a slope of [tex]\(-\frac{3}{2}\)[/tex] and a y-intercept of [tex]\(-7\)[/tex].
- For the inequality [tex]\( y < \frac{1}{2} x + 1 \)[/tex], we first draw the line [tex]\( y = \frac{1}{2} x + 1 \)[/tex]. This line has a slope of [tex]\(\frac{1}{2}\)[/tex] and a y-intercept of [tex]\(1\)[/tex].

2. Determine the region for each inequality:
- The region below the line [tex]\( y = -\frac{3}{2} x - 7 \)[/tex] represents the values where [tex]\( y < -\frac{3}{2} x - 7 \)[/tex].
- The region below the line [tex]\( y = \frac{1}{2} x + 1 \)[/tex] represents the values where [tex]\( y < \frac{1}{2} x + 1 \)[/tex].

3. Find the intersection point:
- To find where these two lines intersect, solve the equations simultaneously:
[tex]\[ -\frac{3}{2}x - 7 = \frac{1}{2}x + 1 \][/tex]
[tex]\[ -\frac{3}{2}x - \frac{1}{2}x = 1 + 7 \][/tex]
[tex]\[ -2x = 8 \][/tex]
[tex]\[ x = -4 \][/tex]

Substitute [tex]\( x = -4 \)[/tex] into one of the equations to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{2}(-4) + 1 \][/tex]
[tex]\[ y = -2 + 1 \][/tex]
[tex]\[ y = -1 \][/tex]

So, the intersection point is [tex]\((-4, -1)\)[/tex].

4. Identify a point in the solution region:
- Now we should identify a point that lies below both lines. A point in the solution set must satisfy both inequalities. A good choice can be slightly below and to the left of the intersection point [tex]\((-4, -1)\)[/tex].

Let's choose the point [tex]\((-5, -2)\)[/tex].

5. Verify the point:
- Check to ensure [tex]\((-5, -2)\)[/tex] satisfies both inequalities:
- For [tex]\( y < -\frac{3}{2} x - 7 \)[/tex]:
[tex]\[ -2 < -\frac{3}{2}(-5) - 7 \][/tex]
[tex]\[ -2 < 7.5 - 7 \][/tex]
[tex]\[ -2 < 0.5 \quad \text{(True)} \][/tex]

- For [tex]\( y < \frac{1}{2} x + 1 \)[/tex]:
[tex]\[ -2 < \frac{1}{2}(-5) + 1 \][/tex]
[tex]\[ -2 < -2.5 + 1 \][/tex]
[tex]\[ -2 < -1.5 \quad \text{(True)} \][/tex]

Since the point [tex]\((-5, -2)\)[/tex] satisfies both inequalities, it is in the solution set.

Thus, the coordinates of a point in the solution set are:
[tex]\[ \boxed{(-5, -2)} \][/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.