bExample Solution: 10 grams of silver and 3 grams of gold will make a ring that will have a mass of 13 grams and cost $79.
Practice makes perfect
a We are given information about a jeweler creating a ring. Let's start by assigning variables.
Grams of silver used: x
Grams of gold used: y
We are also given three different constraints for the mass and cost of the ring. We can create inequalities from that information.
The mass must be more than 10 grams but less than 20 grams
There must be at least 2 grams of gold used
The total cost must be less than $ 90
10 < x+ y < 20
y ≥ 2
0.4 x+25 y < 90
Now that we have the inequalities written we need to graph them. Since one of the inequalities is a compound inequality, we will break that into two simple inequalities. We now have the following system.
10 < x+ y & (I) x+ y < 20 & (II) y ≥ 2 & (III) 0.4 x+25 y < 90 & (IV)
Let's start by creating the boundary line equations by isolating the y-variable for each of the equations.
10 = x+ y x+ y = 20 y = 2 0.4 x+25 y = 90 ⇒
y=- x+10 y=- x+20 y = 2 y=- 2125 x+ 185
We now have the equations for all of the boundary lines. Let's graph them. We will start with the first equation. Since the inequality is less than the line will be dashed. When graphing we must keep in mind that neither x nor y can have negative values.
Now we will use a test point, ( 0, 0), to find out how to shade.
Since the test point made the inequality false we will shade up and away from the point.
We will follow the same process to graph the other inequalities. We have already found the boundary lines. We will use the same test point to determine where to shade.
Inequality
Solid or Dashed Line
True or False
x+ y<20
Dashed
0+ 0<20 True
y≥2
Solid
0≱2 False
0.4 x+25 y<90
Dashed
0.4( 0)+25( 0)<90 True
Now we can graph all of the inequalities and shade them appropriately.
The solution region is where all of the shading overlaps.
b In order to find the mass and cost of one of the solutions, we need to choose a point in the shaded region.
We will use the point ( 10, 3). Note that this is just one point out of many that can be used. We need to use the expressions for the cost and the mass to find the correct values.
Mass:& x+ y
Cost:& 0.4 x+25 y
We can substitute the values into the first expression to find the mass.
