Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

Question: Solve the following linear inequalities in two variables using the graphing method.

1. [tex]\[
\begin{cases}
y \geq -\frac{1}{2}x - 2 \\
y \leq x + 4 \\
x \ \textless \ 4
\end{cases}
\][/tex]

2. [tex]\[
\begin{cases}
y \ \textgreater \ -\frac{3}{2}x - 6 \\
y \geq \frac{3}{2}x \\
y \ \textless \ 5
\end{cases}
\][/tex]


Sagot :

# Solving Linear Inequalities in Two Variables using the Graph Method

## Part A

### Inequalities:
1. [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex]
2. [tex]\(y \leq x + 4\)[/tex]
3. [tex]\(x < 4\)[/tex]

### Steps to Solve:

1. Graph the first inequality [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex]:
- Plot the line [tex]\(y = -\frac{1}{2}x - 2\)[/tex]. Since the inequality is [tex]\(\geq\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\(y = -\frac{1}{2}(4) - 2 = -4\)[/tex].

2. Graph the second inequality [tex]\(y \leq x + 4\)[/tex]:
- Plot the line [tex]\(y = x + 4\)[/tex]. Since the inequality is [tex]\(\leq\)[/tex], shade the region below this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex].
- When [tex]\(x = -4\)[/tex], [tex]\(y = 0\)[/tex].

3. Graph the third inequality [tex]\(x < 4\)[/tex]:
- Draw the vertical line [tex]\(x = 4\)[/tex]. Since the inequality is [tex]\(<\)[/tex], shade the region to the left of this line.

4. Find the intersection of the shaded regions:
- The solution to this system is the region where all three shaded areas overlap.

### Solution:
- The solution is the region that is:
- Above the line [tex]\(y = -\frac{1}{2}x - 2\)[/tex]
- Below the line [tex]\(y = x + 4\)[/tex]
- To the left of the line [tex]\(x = 4\)[/tex].

### Visual Representation:
- The region of overlap can be seen on the graph by shading the corresponding areas accordingly. The solution is the triangular region where the conditions [tex]\(x < 4\)[/tex], [tex]\(y \geq -\frac{1}{2}x - 2\)[/tex], and [tex]\(y \leq x + 4\)[/tex] are satisfied.

## Part B

### Inequalities:
1. [tex]\(y > -\frac{3}{2}x - 6\)[/tex]
2. [tex]\(y \geq \frac{3}{2}x\)[/tex]
3. [tex]\(y < 5\)[/tex]

### Steps to Solve:

1. Graph the first inequality [tex]\(y > -\frac{3}{2}x - 6\)[/tex]:
- Plot the line [tex]\(y = -\frac{3}{2}x - 6\)[/tex]. Since the inequality is [tex]\(>\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -6\)[/tex].
- When [tex]\(x = -4\)[/tex], [tex]\(y = 0\)[/tex].

2. Graph the second inequality [tex]\(y \geq \frac{3}{2}x\)[/tex]:
- Plot the line [tex]\(y = \frac{3}{2}x\)[/tex]. Since the inequality is [tex]\(\geq\)[/tex], shade the region above this line.
- Points to plot the line:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 0\)[/tex].
- When [tex]\(x = 4\)[/tex], [tex]\(y = 6\)[/tex].

3. Graph the third inequality [tex]\(y < 5\)[/tex]:
- Draw the horizontal line [tex]\(y = 5\)[/tex]. Since the inequality is [tex]\(<\)[/tex], shade the region below this line.

4. Find the intersection of the shaded regions:
- The solution to this system is the region where all three shaded areas overlap.

### Solution:
- The solution is the region that is:
- Above the line [tex]\(y = -\frac{3}{2}x - 6\)[/tex]
- Above the line [tex]\(y = \frac{3}{2}x\)[/tex]
- Below the line [tex]\(y = 5\)[/tex].

### Visual Representation:
- The region of overlap can be seen on the graph by shading the corresponding areas accordingly. The solution is the area bounded by [tex]\(y > -\frac{3}{2}x - 6\)[/tex], [tex]\(y \geq \frac{3}{2}x\)[/tex], and [tex]\(y < 5\)[/tex], forming a polygonal region.

In both parts, by plotting the lines and shading the appropriate regions, the solution to the system of inequalities is the area where all conditions are satisfied simultaneously. A graphing calculator or software can assist in visually verifying the solution.