An exam consists of two parts, Section X and Section Y. There can be a maximum of 95 questions. There must be at least 20 more questions in Section Y than in Section X. Write a system of inequalities to model the number of questions in each of the two sections. Then solve the system by graphing.
Algebra 2 Semester A Review
1.) -204.8
2.) 15-gallon tank
3.) 9.7
4.) independent
5.) B
Y >_ (more than or equal to sign) +20
6.) -2 -7|3
7 -6|12
7.) Reflect across the x-axis; translate 3 units to the right; translate up two units
8.) y=-2.5x^2+4.5x-4.5
9.) D
10.) A
11.) 1,-1,6i,-6i
12.) y=641(1/2)^1/68x;615.390 kg
13.) 3.905
14.) 7/12
15.) Associative Property of Multiplication
16.) y= -3/8x-5/4;
slope: -3/8; y-intercept: -5/4
AKA B
x+y <= 95
y >= x+20
Alright, let me clown around with this question a bit!
Let's call the number of questions in Section X "X" and the number of questions in Section Y "Y."
According to the given conditions, we can write the following system of inequalities:
1. X + Y ≤ 95 (There can be a maximum of 95 questions in total.)
2. Y ≥ X + 20 (There must be at least 20 more questions in Section Y than in Section X.)
To solve this system by graphing, we can start by converting the inequalities into equations:
1. X + Y = 95
2. Y = X + 20
Now, let's plug in some numbers and create a laughing graph!
If we set X = 0, equation 2 will give us Y = 20. If we set Y = 0, equation 2 will give us X = -20, but since we can't have a negative number of questions, we'll ignore this point.
Let's plot these points, shall we? (Imagine an x-y graph with axes and a clown juggling multiple balls.)
(X, Y) = (0, 20)
Now we can draw a line connecting this point to the point (95, 0). The line would look like the clown's balancing act!
Lastly, since the inequalities require X + Y to be less than or equal to 95, we will lightly shade the area below the line, resembling the clown's shadow on the ground.
Voila! We've created a jocular visual representation of the solution to this system of inequalities.
Let's define the variables as follows:
- Let X represent the number of questions in Section X.
- Let Y represent the number of questions in Section Y.
According to the given information, we can write the following inequalities:
1. X + Y ≤ 95
This inequality states that the total number of questions in both sections combined cannot exceed 95.
2. Y ≥ X + 20
This inequality ensures that there are at least 20 more questions in Section Y than in Section X.
To solve this system by graphing, we'll plot these two inequalities on a graph:
First, let's graph the equation X + Y = 95 as a solid line:
- Plot the points (0, 95) and (95, 0).
- Draw a line passing through these two points.
Next, let's graph the equation Y = X + 20 as a dashed line:
- Plot the point (0, 20).
- Using a slope of 1, move up 1 unit and right 1 unit to plot another point.
- Repeat this process until you have enough points to draw a dashed line.
Finally, we need to shade the region that satisfies the system of inequalities:
- The shaded region is the area below and including the solid line (X + Y ≤ 95).
- The shaded region is also the area above and including the dashed line (Y ≥ X + 20).
The solution to this system of inequalities is the region where the shaded areas overlap.
Note: The graph is approximate, and it's recommended to use graphing software or a graphing calculator for a more accurate representation.
To write a system of inequalities, let's first define our variables. Let's say the number of questions in Section X is denoted by the variable "x," and the number of questions in Section Y is denoted by the variable "y."
Based on the given constraints, we can write the following system of inequalities:
1. The total number of questions in both sections cannot exceed 95:
x + y ≤ 95
2. There must be at least 20 more questions in Section Y than in Section X:
y ≥ x + 20
Now, let's solve this system by graphing.
To do so, we'll graph the boundary lines for both inequalities and shade the feasible region that satisfies both conditions.
First, let's graph the equation x + y ≤ 95 by drawing the line x + y = 95. To graph it, we'll plot two points on the line and connect them.
When x = 0, we have y = 95. So one point on the line is (0, 95).
When y = 0, we have x = 95. So another point on the line is (95, 0).
Now, let's graph the equation y ≥ x + 20. To do that, we'll graph the line x + 20 = y. Again, we'll plot two points on this line and connect them.
When x = 0, we have y = 20. So one point on the line is (0, 20).
When y = 0, we have x = -20. So another point on the line is (-20, 0).
Now, we'll shade the region that satisfies both inequalities. Since we want y to be greater than or equal to (x + 20), we'll shade the area above the line x + 20 = y.
The shaded region where the two lines overlap represents the feasible region that satisfies both conditions, which gives us the possible solutions to our system of inequalities.
Now, you can graph the lines and shade the region to solve the system visually.