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
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.
To find the ordered pair that is included in the solution to this system of inequalities, we need to solve the system of equations created by the given inequalities.
Step 1: Write the system of equations:
y ≤ (2/3)x + 1
y > (-1/4)x + 2
Step 2: Solve the system of equations:
We can solve these inequalities simultaneously by graphing them or using algebraic methods. To use algebraic methods, we can solve the system by substituting one equation into the other.
Let's solve both inequalities step-by-step:
For the first inequality (y ≤ (2/3)x + 1):
We can graph this equation by plotting two points and drawing a line through them, but we can also solve it algebraically.
- First, subtract (2/3)x from both sides:
y - (2/3)x ≤ 1
- To isolate y, add (2/3)x to both sides:
y ≤ (2/3)x + 1
So, this inequality represents a line with a slope of (2/3) and a y-intercept of 1. The shaded area below or on this line represents the solutions to this inequality.
For the second inequality (y > (-1/4)x + 2):
We can solve this equation algebraically as well.
- First, subtract (-1/4)x from both sides:
y - (-1/4)x > 2
- Simplify:
y + (1/4)x > 2
- To isolate y, subtract (1/4)x from both sides:
y > (-1/4)x + 2
So, this inequality represents a line with a slope of (-1/4) and a y-intercept of 2. The shaded area above this line represents the solutions to this inequality.
Step 3: Find the common shaded region:
To find the ordered pair that satisfies both inequalities, we need to find the overlapping region between the two shaded areas.
Upon analyzing the two lines, we can see that they intersect. Let's solve the system of equations to find the point of intersection, which will be our common solution:
(2/3)x + 1 = (-1/4)x + 2
- First, add (1/4)x to both sides:
(2/3 + 1/4)x = 2 - 1
- Simplify:
(11/12)x = 1
- Now, isolate x by multiplying both sides by 12/11:
x = (12/11)
- Substitute this value back into one of the original equations to find y:
y = (2/3)(12/11) + 1
y = 24/33 + 1
y = 8/11 + 24/33
y = (72/33) + (24/33)
y = 96/33
y = 32/11
So, the ordered pair that is included in the solution to this system of inequalities is (12/11, 32/11).