We can solve this system of inequalities by graphing them on the coordinate plane and finding the overlapping region. Alternatively, we can manipulate the inequalities to isolate y on one side and then compare the coefficients of x to see which inequality has the smaller slope.
For the first inequality, we have:
y ≤ (2/3)x + 1
Subtracting 1 from both sides, we get:
y - 1 ≤ (2/3)x
Multiplying both sides by 3/2 (which is the reciprocal of 2/3), we get:
(3/2)(y - 1) ≤ x
Simplifying, we get:
x ≥ (3/2)y - 3/2
For the second inequality, we have:
y > (-1/4)x + 2
Adding 1/4x to both sides, we get:
y + (1/4)x > 2
Multiplying both sides by 4/1 (which is the reciprocal of 1/4), we get:
4y + x > 8
So the system of inequalities can be rewritten as:
x ≥ (3/2)y - 3/2
4y + x > 8
To find which ordered pair is included in the solution, we can choose a point on one side of each inequality and plug it into the inequalities to see which ones are true. For example, if we choose the point (2,1), we get:
(2,1) satisfies the first inequality because 1 ≤ (2/3)(2) + 1 is true.
(2,1) does not satisfy the second inequality because 4(1) + 2 is not greater than 8.
Therefore, the ordered pair (2,1) is not included in the solution to the system of inequalities. We can repeat this process with other points or use the equations we derived to graph the inequalities and find the overlapping region.
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
1 answer
1 answer