To solve linear inequalities in two variables using the graphing method, you should graph each inequality on the same set of coordinate axes and identify the region that satisfies all the inequalities simultaneously.
a. The given system of inequalities is:
[tex]\[
\begin{cases}
y > -\frac{1}{2}x - 2 \\
y \leq x + 4 \\
x < 4
\end{cases}
\][/tex]
1. Graph the inequality [tex]\( y > -\frac{1}{2}x - 2 \)[/tex]:
- First, graph the line [tex]\( y = -\frac{1}{2}x - 2 \)[/tex]. This line has a slope of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept at [tex]\(-2\)[/tex].
- To graph this, plot the y-intercept at (0, -2) and use the slope to plot another point. From (0, -2), move down 1 unit and right 2 units to reach another point at (2, -3).
- Since the inequality is [tex]\( y > -\frac{1}{2}x - 2 \)[/tex], you will shade the region above this line.
- Use a dashed line for [tex]\( y = -\frac{1}{2}x - 2 \)[/tex] to indicate that points on the line itself are not included in the solution.
2. Graph the inequality [tex]\( y \leq x + 4 \)[/tex]:
- Graph the line [tex]\( y = x + 4 \)[/tex]. This line has a slope of 1 and a y-intercept at 4.
- To graph this, plot the y-intercept at (0, 4) and use the slope to plot another point. From (0, 4), move up 1 unit and right 1 unit to reach another point at (1, 5).
- Since the inequality is [tex]\( y \leq x + 4 \)[/tex], you will shade the region below or on this line.
- Use a solid line for [tex]\( y = x + 4 \)[/tex] to indicate that points on the line are included in the solution.
3. Graph the inequality [tex]\( x < 4 \)[/tex]:
- Graph the vertical line [tex]\( x = 4 \)[/tex].
- Since the inequality is [tex]\( x < 4 \)[/tex], you will shade the region to the left of this line.
- Use a dashed line for [tex]\( x = 4 \)[/tex] to indicate that points on the line itself are not included in the solution.
Now, combine the shaded regions from all three inequalities. The solution to the system is the region where the shaded areas of the three inequalities overlap.
b. The given system of inequalities is:
[tex]\[
\begin{cases}
y > -\frac{3}{2}x - 6 \\
y \geq \frac{3}{2}x \\
y < 5^2
\end{cases}
\][/tex]
1. Graph the inequality [tex]\( y > -\frac{3}{2}x - 6 \)[/tex]:
- Graph the line [tex]\( y = -\frac{3}{2}x - 6 \)[/tex]. This line has a slope of [tex]\(-\frac{3}{2}\)[/tex] and a y-intercept at -6.
- To graph this, plot the y-intercept at (0, -6) and use the slope to plot another point. From (0, -6), move down 3 units and right 2 units to reach another point at (2, -9).
- Since the inequality is [tex]\( y > -\frac{3}{2}x - 6 \)[/tex], shade the region above this line.
- Use a dashed line for [tex]\( y = -\frac{3}{2}x - 6 \)[/tex] to indicate that points on the line itself are not included in the solution.
2. Graph the inequality [tex]\( y \geq \frac{3}{2}x \)[/tex]:
- Graph the line [tex]\( y = \frac{3}{2}x \)[/tex]. This line has a slope of [tex]\(\frac{3}{2}\)[/tex] and passes through the origin (0, 0).
- Since the inequality is [tex]\( y \geq \frac{3}{2}x \)[/tex], you will shade the region above or on this line.
- Use a solid line for [tex]\( y = \frac{3}{2}x \)[/tex] to indicate that points on the line are included in the solution.
3. Graph the inequality [tex]\( y < 25 \)[/tex]:
- Graph the horizontal line [tex]\( y = 25 \)[/tex].
- Since the inequality is [tex]\( y < 25 \)[/tex], you will shade the region below this line.
- Use a dashed line for [tex]\( y = 25 \)[/tex] to indicate that points on the line itself are not included in the solution.
Now, combine the shaded regions from all three inequalities. The solution to this system is the region where the shaded areas of the three inequalities overlap.
