An exam consists of two parts, Section X and Section Y. There can be a maximum of 80 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.
X + Y =< 80.
Y => x+20.
X + (X+20) =< 80.
2x+ 20 =< 80.
2x =< 80-20.
3x =< 60.
X =< 20.
Y => 20 + 20.
Y => 40.
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Let's denote the number of questions in Section X as "x" and the number of questions in Section Y as "y".
According to the given conditions, we can write the following system of inequalities:
1. There can be a maximum of 80 questions:
x + y ≤ 80
2. There must be at least 20 more questions in Section Y than in Section X:
y ≥ x + 20
To solve the system graphically, we can plot these inequalities on a coordinate plane.
First, let's graph the equation x + y = 80:
To do this, we'll plot two points (0,80) and (80,0), and draw a line through them.
Next, let's graph the inequality y ≥ x + 20:
To do this, we'll start by graphing the line y = x + 20.
We'll plot two points (0,20) and (60,80), and draw a dashed line that passes through them. Note that this line is dashed because the inequality symbol is ≥ (greater than or equal to).
Finally, we'll shade the region above the line y = x + 20, as this represents the values that satisfy the inequality y ≥ x + 20.
The shaded region will be bounded by the line x + y = 80 and the line y = x + 20.
To model the number of questions in each section, let's represent the number of questions in Section X as "x" and the number of questions in Section Y as "y".
According to the given conditions:
1) The total number of questions must be a maximum of 80: x + y ≤ 80
This inequality ensures that the sum of questions in both sections does not exceed 80.
2) There should be at least 20 more questions in Section Y than in Section X: y ≥ x + 20
This inequality ensures that the number of questions in Section Y is greater than or equal to the number of questions in Section X plus 20.
Now, let's solve the system of inequalities by graphing:
First, graph the equation x + y ≤ 80:
Plot the line x + y = 80 and shade the area below the line. Since the inequality is ≤, the line should be a solid line.
Next, graph the equation y ≥ x + 20:
Plot the line y = x + 20 and shade the area above the line. Since the inequality is ≥, the line should be a dashed line.
The solution to the system of inequalities will be the overlapping shaded region. The points within this region represent valid combinations of (x, y) that satisfy the given conditions.
By examining the graph, you can identify different coordinate points within the overlapping region to find the possible numbers of questions in each section that satisfy the conditions.