Liam was selling DVDs. The prices of the DVDs can be described by the equation the absolute value of 𝑐 minus 15 equals four where 𝑐 is the price in dollars. By graphing the equation 𝑦 equals the absolute value of 𝑥 minus 15 or otherwise, determine the highest and lowest cost of a DVD.
The cost of Liam’s DVDs can be described by the absolute value of 𝑐 minus 15 is equal to four. This question is suggesting that one method for solving is graphing the equation 𝑦 equals the absolute value of 𝑥 minus 15. To solve by graphing, we’ll first sketch a coordinate grid. We’ll only be dealing with quadrant one as 𝑥 is the cost of the DVD, and that needs to be positive. And the output 𝑦 is an output of an absolute value which is also positive.
To graph the function 𝑦 equals the absolute value of 𝑥 minus 15, we could select some values for 𝑥 and solve for 𝑦. When 𝑥 equals zero, 𝑦 is equal to the absolute value of zero minus 15. The absolute value of negative 15 is positive 15. And so, we could graph the point zero, 15. If we select 𝑥 equal to five, we need to solve the absolute value of five minus 15, which will be the absolute value of negative 10 which is positive 10 and gives us the coordinates five, 10.
From there, when 𝑥 equals 10, 𝑦 will be equal to five. When 𝑥 equals 15, 𝑦 equals zero. When 𝑥 equals 20, 𝑦 equals five, which we can graph. And then, we’re starting to see that a symmetrical V shape of the absolute value functions. And we complete our sketch of this graph by connecting the points we have plotted. And we say that this is the sketch of the function 𝑦 equals the absolute value of 𝑥 minus 15.
At this point, it might not be clear how this helps us solve the prices of DVDs for Liam. And this is when we need to look carefully at Liam’s equation. Liam says the cost of his DVDs can be found by taking the absolute value of 𝑐 minus 15 and setting that equal to four. If four is equal to the absolute value of 𝑐 minus 15, then on our graph we’re interested in the places where 𝑦 is equal to four. We can graph a second line that is 𝑦 equals four. And our solutions will be found at the intersection of these two lines. We have an intersection at the point 11, four and an intersection at the point 19, four. We found two 𝑐-values for Liam. That would be 11 and 19.
Solving a question like this with the graph requires you to be very accurate with your sketches. So, let’s consider how we would solve this algebraically and confirm what we see on our graph. If we know the absolute value of 𝑐 minus 15 equals four, we need to consider the two cases, that is, the case when 𝑐 minus 15 equals four and the negative of that, which we can rewrite as 𝑐 minus 15 equals negative four. To solve for 𝑐 in both cases, we’ll add 15 to both sides of the equation and find that 𝑐 equals 19 or 𝑐 equals 11, which are the input values we found at the intersection points on the graph.
This means that Liam’s highest-costing DVD is 19 dollars and his lowest-costing DVD is 11 dollars.