Solving Systems of Inequalities

Linear Programming

Linear programming is used to find the maximum or minimum value of a function that is subject to a set of constraints.

Linear programming is a process for finding a maximum or minimum value of a linear function when there are restrictions involved. The function that is maximized or minimized is called the objective function. A constraint is an inequality that represents a restriction of the objective function. The set of constraints are modeled by a system of linear inequalities.

The intersection of the constraints is called the feasible region. The maximum and minimum values of the objective function must occur at a corner point, or vertex, of the feasible region.

To solve a linear programming problem:

1. Use the information in the problem to write the constraints as a system of linear inequalities. Note that the constraints x0x \geq 0 and y0y \geq 0 may not be stated but are often implied in the problem.

2. Graph the system of linear inequalities, and find the coordinates of the corner points of the feasible region.

3. Use the information in the problem to write the objective function.

4. Evaluate the objective function at each of the corner points to identify the maximum or minimum value.

Step-By-Step Example
Solving a Linear Programming Problem
A jeweler is making necklaces and bracelets to sell at a festival. She has up to 36 hours to work on the jewelry. Each necklace takes 1.5 hours to make, and each bracelet takes 0.75 hours to make. She can spend up to $90 on supplies. The supplies for a necklace cost $3, and the supplies for a bracelet cost $2. Her profit for selling each necklace is $22, and her profit for selling each bracelet is $14 . How many necklaces and bracelets should she make to maximize her profit?
Step 1

Write a system of inequalities made up of the constraints. Let xx represent the number of necklaces and yy represent the number of bracelets the jeweler makes.

Write one inequality for her time: Each necklace takes 1.5 hours to make, and each bracelet takes 0.75 hours to make. The total time she spends making jewelry can be up to 36 hours.
1.5x+0.75y361.5x+0.75y\leq 36
Solve this inequality for yy .
y2x+48y\leq -2x+48
Write another inequality for the costs: The supplies for a necklace cost $3, and the supplies for a bracelet cost $2. The total amount she spends on supplies can be up to $90.
3x+2y903x+2y\leq 90
Solve this inequality for yy .
y32x+45y\leq -\frac{3}{2}x+45
She cannot make a negative number of necklaces or bracelets, so x0x \geq 0 and y0y \geq 0. Write the system of inequalities.
{x0y0y2x+48y32x+45{\begin{cases}x & \geq &0\hspace{35pt} \\ y & \geq &0\hspace{35pt} \\ y & \leq & -2x+48 \\ y & \leq & -\frac{3}{2}x + 45 \end{cases}}
Step 2

Graph the system of linear inequalities, and find the coordinates of the corner points of the feasible region.

The constraints x0x \geq 0 and y0y\geq 0 mean that the graph is restricted to the first quadrant.

Examine the inequality symbols in the remaining constraints:
y2x+48y32x+45\begin{aligned}y&\leq -2x+48\\y&\leq -\frac{3}{2}x+45\end{aligned}
The inequality symbols are \leq, so the lines are both solid and the shaded area is below each line.
Two of the corner points are intercepts of the lines, and one is the origin. The fourth corner point is the intersection of the constraints that are not the xx- and yy-axes. The coordinates of this point are (6,36)(6, 36).

The four corner points are (0,0)(0,0), (24,0)(24, 0), (6,36)(6,36), and (0,45)(0,45).

Step 3

Use the information in the problem to write the objective function.

The jeweler's profit for selling each necklace is $22, and her profit for selling each bracelet is $14. The function that represents her profit PP for selling xx necklaces and yy bracelets is:
P=22x+14yP=22x+14y
Step 4
Evaluate the objective function at each of the four corner points, (0,0)(0,0), (24,0)(24, 0), (6,36)(6,36), and (0,45)(0,45).
(0,0):P=22(0)+14(0)=0(24,0):P=22(24)+14(0)=528(6,36):P=22(6)+14(36)=636(0,45):P=22(0)+14(45)=630\begin{aligned}&(0,0)\text{:}\; P=22(0)+14(0)=0\\&(24, 0)\text{:}\;P=22(24)+14(0)=528\\&(6,36)\text{:} \;P=22(6)+14(36)=636\\&(0,45)\text{:} \;P=22(0)+14(45)=630\end{aligned}
Solution

Maximizing the objective function will maximize the jeweler's profit. To maximize the objective function, choose the corner point that results in the greatest value.

The point (6,36)(6, 36) results in the greatest value for the profit function, so the jeweler should make 6 necklaces and 36 bracelets. Her profit will be $636.