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$
$10 < x + y < 20$	$y \geq 2$	$0.4 x + 25 y < 90$

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 x + y < 20 y \geq 2 0.4 x + 25 y < 90 (I) (II) (III) (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 \Rightarrow ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎧ y = - x + 10 y = - x + 20 y = 2 y = - \frac{2}{1 2 5} x + \frac{1 8}{5}

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.

10 < x + y

SubstituteII

$x = 0$ , $y = 0$

10 < ? 0 + 0

AddTerms

Add terms

10 ≮ 0

Since the test point made the inequality false we will shade up and away from the point.

Diagram showing the solutions to the inequality y greater than -x+10

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
$y \geq 2$	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.

Diagram showing the solutions to each of the four inequalities overlapping each other

The solution region is where all of the shading overlaps.

Diagram showing the solution set to the system of inequalities

Exercise