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]



Answer :

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]