Hi. This is Anthony Varela. and today we're going to be using linear inequalities in real world scenarios. So we're going to be developing inequalities to represent some constraints, then we're going to construct a graph, and then we'll use the graph or the inequalities to solve some problems related to this scenario.
So here is our situation related to a budget restriction. So an organization is putting together an event, and they have two main costs. They're going to hire a band, so they have a flat fee of $500 for the event. And they're also going to be serving food. And the catering company has said it's going to cost $8 for every person that comes.
So we can then create an expression to represent this cost. So this cost is going to be 500 plus 8 times x. So here, x equals the number of people that come. Now this is an expression for the cost. Now thinking about this company or this organization, they have a budget, and this budget must be, then, greater than the cost so they can afford it. So we're going to say that y equals the budget. And in order to afford the music and the food, their budget has to be greater than or equal to that cost so that they have enough money to pay the bills.
So we're using this inequality symbol then, greater than or equal to, because the budget could certainly equal the exact cost. That would be OK. The budget can certainly be greater than the cost, then they'll have money left over. But the budget cannot be underneath that cost because then they won't have enough money to pay for music and the food.
So now we're going to go ahead and construct a graph of this situation. So we're going to graph this inequality, y is greater than or equal to 500 plus 8x. So the first thing that I need to be thinking about is, what kind of line am I going to use to graph? Now if my inequality symbols are less than or greater than, I'm going to be using a dashed line. If my inequality symbols include or equal to, I'm going to be using a solid line. So I'll be using a solid line because my inequality symbol allows for exactly equal to.
So this inequality resembles an equation in slope-intercept form. If this is y equals 500 plus 8x, I'd be able to graph a line looking at slope and looking at the intercept. So here would be, then, the intercept, 500. So I'm going to find that, then, on my graph, so here's 500. And thinking about the slope of 8x, now that would mean I have a rise of 8 for every run of 1. And looking at my scale, that's a little bit hard to see, so I'm going to multiply this out a bit.
If my run, let's say, was 50-- so if we went over 50-- my rise would have to be 50 times eight. Now that would be 400. So my rise would go up, let's see, 400. So 100, 200, 300, 400. So this is going to be another point on that line. So here, then, is the line that's y equals 500 plus 8x.
Now I need to be thinking about y is greater than or equal to 500 plus 8x. Now before I actually go ahead and shade in a region, I'd like to talk about how real world scenarios often have a natural restriction of another type. Thinking about x being the number of people that come to this event, it doesn't make sense, realistically speaking, to have a negative attendance. So we actually have a system of inequalities here that x is greater than or equal to zero. So we're only going to be looking at the positive values of x because we can't have negative people coming.
So I'm going to shade in this y is greater than or equal to 500 plus x but I'm only going to do it on the positive x side. So this is really going to be shaded solution to a system of equations. So looking at my inequality symbol here, this is greater than or equal to, so I'm going to be shading above. So here is, then, my solution region for this system.
Now I'm going to call this a feasibility region. This is another word for solution region, but I like feasibility region in this case because it's going to really make sense when we're talking about what the solution region means. So let's pick a point that fits within this feasibility region. So we have the point 50, 1,000. So this is an x and y-coordinate. Remember, x is people, and y is budget. So what this point means is that it's feasible to host 50 people with a budget of $1,000.
So I'm going to take this inequality y is greater than 500 plus 8x. I'm going to plug in 50 for x and 1,000 for y. And we're going to show that this is a true statement because it fits within our feasibility region. So eight times 50 is 400, and then when I add that to 500, I get 900. So I have this inequality statement that 1,000 is greater than or equal to 900, and that's true. It fits within our feasibility region. This means that if 50 people come and we only have a budget of $1,000, we can make that happen. That's feasible.
So let's pick a point that appears to be just on this edge here. It looks like it's not in our feasibility region. This is 80 people and $1,100 as our budget. And let's plug this in to y is greater than or equal to 500 plus 8x. So eight times 80 is 640. And when I add that to 500, I get 1,140. And we see that this statement of inequality is not true. 1,100 is not greater than or equal to 1,140. So this fits outside of our feasibility region. And what this means is that if we have 80 people come and we only have $1,100 in our budget, this is not going to work. That's not feasible. We need a bigger budget to host 80 people.
So a feasibility region can also be thought of that solution region. And I think feasibility really makes sense when we're talking about budget and certain constraints. So we're going to be answering a couple of questions involving this situation. And our first question is, is it feasible to host 140 people with a budget of $1,700? So looking at my graph right here, locating 140 on the x-axis, I don't have it graphed here so I'm not exactly sure if I can justify just looking at the graph if this is feasible or not. So I'm just going to plug in, then, 140 for x and 1,700 for y, and see if I get a true statement.
So 8 times 140 is 1,120. And when I add that to 500, I get 1,620. So 1,700 is greater than or equal to 1,620-- not by much but it does work. So it is feasible to host 140 people with a budget of $1,700.
Our next question is, how many people could attend with a budget of $1,350? So looking at $1,350 on my graph, so here's 1,000, 1,100, 1,200, 1,300. So 1,350 is right here. So I can go right up to the edge of my line, So. My answer is going to be somewhere in between 100 and 200. But let's find the exact value.
So I'm going to plug in 1350 for y, and I need to solve for x. How many people can attend? So I'm going to subtract 500 from both sides of my inequality. So 850 is greater than or equal to 8x. Dividing both sides by 8, I have that 106.25 is greater than or equal to x. Now thinking about what x means, x represents people, and we can't really have one fourth of a person. So realistically speaking, this means at most 106 people can come. Because if we had 107 people come, that would be then over our budget.
So let's review using linear inequalities in real world scenarios. We developed an inequality using x equals people and y equals budget for an organization putting together an event. And our inequalities that we developed were y is greater than or equal to 500 plus 8x. 500 was the cost of the music, and 8x was the cost of food per person. Now we also had this realistic boundary that it doesn't make sense to have negative people, so x was greater than or equal to 0.
When we were graphing this, if we had our strict inequality symbols, we would use a dashed line. If we have our non-strict inequality symbols, we'd use a solid line. And then we shaded in regions of the coordinate plane, and we called that a feasibility region or solution region that represented people that could come to our events given our budget. Was it feasible or not? So thanks for watching this tutorial on using linear inequalities in real world scenarios. Hope to see you next time.
