Writing a System from a Graph
Consider the graph below:
In order to write the inequalities that make up this system, we need to look at each individual piece to the graph and construct an inequality. We'll focus on the blue inequality first:
The first thing to note is that the boundary line is a dashed line. This means that the inequality symbol will be either < or > as it is strict and does not include the exact points on the boundary line. We can also note the arrows coming from the boundary line that show that all coordinate points above the boundary line satisfy the inequality. This means that the inequality symbol is either > or ≥. Since we have already determined the inequality symbol is strict, we know that the symbol we will use for the blue line is > or the "greater than" symbol.
Now, we can think of the boundary line as an equation, and write an equation in the form y = mx + b, and simply replace the equals sign with our inequality symbol. To write the equation in the form y = mx + b, we need to find the slope and the y-intercept.
To find the y-intercept, we look at where the line cross the y-axis. We see that it does so at the point (0, 3), which means our b-value in the equation is 3. Next, we calculate the slope by counting rise and run to get from one point to the next. Lets start at our y-intercept, and count the rise and run to get from the y-intercept to a discernible point on our graph, say (1, 1). The rise is –2 and the run is 1, making our slope –2. The equation of the boundary line is y = -2x + 3
So far, we have determined one inequality in the system:
We follow the same process with the other boundary lines. Let's work with the green line next. We notice a y-intercept of –2. To find the slope, let's count the rise and run from the y-intercept at (0, –2) to another point on the line, say (3, 0). This is a rise of 2 and a run of 3, making our slope 2/3.
Our boundary line is a solid line, narrowing our inequality symbol to either ≤ or ≥. Since the solution region to the system includes all points above this line, we know the inequality symbol is ≥
So far, we have determined two inequalities in our system:
Finally, let's focus on the red boundary line. This line is a horizontal line, so there is actually no x-term in its equation. The boundary line is simply y = 6. To turn this into an inequality, we note the dashed line, and the solution region underneath the line. This means the inequality is y < 6
We have determined all inequalities in the system:
Writing a System from a Scenario
Consider the following scenario:
You manage an art store, and need to order canvas paper and brushes for the studio. You must buy at least 200 sets of paint brushes, which cost $15 each, but no more than 75 rolls of canvas paper, which cost $30 each. Your budget for this expense is $5,000.
We can represent these budget and quantity restrictions with a system of inequalities. The first thing we need to do is define variables. We are purchasing sets of paint brushes and rolls of canvas paper, so we will make these the definitions for x and y:
One of our limitations in our scenario is our budget. We cannot spend more than $5,000 on these supplies. This means that the total cost of supplies must be less than or equal to $5,000 (because our bill can be exactly $5,000 and still fit within our budget). To write this inequality, we will express the total cost for paint brush sets and canvas paper rolls, and restrict it to less than or equal to $5,000:
Note that to express cost, we multiplied x and y by their respective costs: $15 for each paint brush set, and $30 for each roll of canvas paper.
We also have additional restrictions from our scenario. These restrictions involve quantity: we must buy at least 200 sets of brushes, and no more than 75 rolls of canvas paper. We'll set up a relation between x and 200, as well as y and 75.
"At least" means that we can include the exact value of the limitation, so our second inequality in the system is
This means that buying 200 paint brush sets is okay. Even buying 300 sets of paint brushes meets this restriction (although it does put our other restrictions at risk).
"No more than" also means that we can include the exact value of our limitation, so the final inequality in the system is
This means that we are allowed to buy up to 75 rolls of canvas paper, but not 76, or anything higher.
Our system of inequalities for our situation is: