Part (a)
x = number of adults
y = number of students
5x = amount all of the adults paid ($5 each)
2y = amount the students paid ($2 each)
5x+2y = total amount of money for adults and students combined
The total amount (5x+2y) must be more than $150
So we end up with 5x+2y > 150 as the inequality.
This inequality involves the boundary line 5x+2y = 150 which goes through the y and x intercepts of (0,75) and (30,0) respectively. We use a dashed line for the boundary to mean that points on the boundary are not included as solution points.
We shade above the dashed boundary line to finish off the graph.
Refer to figure (a) shown below.
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Part (b)
x = number of adults
y = number of students
x+y = total number of people
120 seats is the max capacity so x+y cannot exceed this value.
x+y = 120 or x+y < 120
Or in short x+y ≤ 120 is the inequality
This involves the boundary line x+y = 120 which goes through the y and x intercepts of (0,120) and (120,0). We shade below the solid boundary line.
Refer to figure (b) shown below.
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Part (c)
Now imagine we overlapped the blue region from part (a) and the red region from part (b)
We'll get the purple region shown in figure (c). At this point, I'm only considering cases when x ≥ 0 and when y ≥ 0. It's not possible to have a negative number of people, so that's why we focus on the upper right quadrant of the xy grid.
This purple region consists of all (x,y) points that make ALL of the inequalities to be true. Exclude any points on the dashed boundary.
For example, if we picked the point (x,y) = (50,40) then this point is in the purple region. We can confirm its a solution by checking each inequality.
5x+2y > 150
5*50+2*40 > 150
250+80 > 150
330 > 150 .... which is true
and,
x+y ≤ 120
50+40 ≤ 120
90 ≤ 120 ... which is also true
Furthermore, both x and y are nonnegative making x ≥ 0 and y ≥ 0 to be true as well. Every inequality mentioned is true for (x,y) = (50,40) to fully confirm the solution point. There are other possible solutions such as (40,60) and (80,20) which I'll let you verify.
If we stick to the (50,40) from earlier, then this means 50 adults and 40 students went to the concert.
Answer: 50 adults and 40 students
Other answers are possible