The following inequalities form a system.
y is less than or equal to two-thirds times x plus 1
y is greater than negative one-fourth times x plus 2
Which ordered pair is included in the solution to this system?
To find the ordered pair(s) included in the solution to this system, we need to solve the system of inequalities.
Given:
1) y ≤ (2/3)x + 1
2) y > (-1/4)x + 2
To find the common solution, we need to identify the region satisfying both inequalities.
Let's start by graphing each inequality.
For equation 1: y ≤ (2/3)x + 1
First, we'll plot the y-intercept at (0,1) and use the slope to draw a line:
Starting at the y-intercept (0,1), we will:
1) Move up 2 units (the numerator of the slope) and right 3 units (the denominator of the slope) to plot the second point.
2) Connect the two points with a solid line since the inequality includes "less than or equal to."
For equation 2: y > (-1/4)x + 2
Again, we'll plot the y-intercept at (0,2) and use the slope to draw a line:
Starting at the y-intercept (0,2), we will:
1) Move down 1 unit (the numerator of the slope) and right 4 units (the denominator of the slope) to plot the second point.
2) Connect the two points with a dotted line since the inequality includes "greater than," but not "equal to."
Now, let's identify the solution region by shading the area that satisfies both inequalities:
Since the inequality symbol in equation 1 includes "less than or equal to," we will shade the region below the line.
Since the inequality symbol in equation 2 includes "greater than," we will shade the region above the line.
The region that satisfies both inequalities is the shaded region where the shaded regions overlap.
Now, let's find the ordered pair(s) within the solution region. To find the common solution, we need to find the point(s) that satisfy both inequalities by examining the shaded region.
Looking at the shaded region, we can see that the point (1, 1) lies within the region that satisfies both inequalities. Therefore, the ordered pair (1, 1) is included in the solution to this system.
So, the ordered pair (1, 1) is included in the solution to this system of inequalities.