Graph the system of inequalities
x (lesser or equal to sign)2
y (greater or equal to sign)-2x+4
I am having trouble understanding what points to plot on the graph and also how the graph should be shaded (if at all).
They are not points.
for x<=2, make a vertical line at x=2
for y>= 2x+4, plot the line y=2x+4. Graph a few points, then connect them in a line.
Now the solution area will be when y is above the slanted line and x is to the right of the vertical line.
To graph the system of inequalities, we will plot the lines represented by each inequality, and then determine the shaded region that satisfies both inequalities.
Let's start with the first inequality:
x ≤ 2
To graph this, we draw a solid vertical line at x = 2, and shade the region to the left (including the line) since x can be equal to 2.
Next, we'll graph the second inequality:
y ≥ -2x + 4
To graph this, we first rewrite it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
y ≥ -2x + 4
The slope of this line is -2, and the y-intercept is 4. So, we plot a point at (0, 4), which is the y-intercept, and from there, we use the slope to find additional points to draw the line.
For example, if we move one unit to the right (increase x by 1), we move two units down (decrease y by 2). So, another point on the line is (1, 2). We can repeat this process to find more points if needed.
Now, we draw a solid line that passes through the plotted points and extends in both directions. Since the original inequality is "greater than or equal to," we will shade the region above the line (including the line) since y can be equal to -2x + 4.
Finally, we need to determine the shaded region that satisfies both inequalities. This region is the intersection of the shaded regions from each inequality. In this case, the shaded region is the region to the left of the vertical line (x ≤ 2) and above the line (y ≥ -2x + 4).
To summarize:
- Plot a solid vertical line at x = 2, and shade the region to the left.
- Plot the line y = -2x + 4, and shade the region above (including) the line.
- The shaded region that satisfies both inequalities is the region to the left of x = 2 and above y = -2x + 4.