Systems of inequalities
An equation in ${x}$ and ${y}$ has a curve or a line as its graph. An inequality, such as ${2x+3y>6}$, has a whole region of the plane as its graph. This page shows how to graph such regions, and how to find the region where several inequalities are all true.
Graphing an inequality
The graph of an inequality in ${x}$ and ${y}$ is the set of all points ${(x,y)}$ that make it true. To graph it:
- Boundary: graph the equation you get by changing the inequality sign to ${=}$. Draw it dashed for ${<}$ or ${>}$ (the boundary is not included), and solid for ${\le}$ or ${\ge}$ (it is included).
- Test point: the boundary splits the plane into regions. Pick a point that is not on the boundary, such as ${(0,0)}$. Put it into the inequality.
- Shade: if the test point makes the inequality true, shade its side of the boundary. If not, shade the other side.
Example 1: Graph the inequality ${2x+3y>6}$.
Solution:
The boundary is the line ${2x+3y=6}$. It crosses the axes at ${(3,0)}$ and ${(0,2)}$. The sign is ${>}$, so draw it dashed.
Test the point ${(0,0)}$: ${2(0)+3(0)=0}$, and ${0>6}$ is false. So shade the side of the line that does not contain the origin.
Example 2: Graph the inequality ${y\ge x^2-2}$.
Solution:
The boundary is the parabola ${y=x^2-2}$. The sign is ${\ge}$, so draw it solid.
Test ${(0,0)}$: ${0\ge 0-2}$ is true. So shade the side that contains the origin: the region inside the parabola.
Watch out: Never use a test point on the boundary itself. If the boundary passes through ${(0,0)}$, pick another point, such as ${(1,0)}$.
Systems of inequalities
The solution set of a system of inequalities is the set of points that make all the inequalities true. On a graph, it is the region where all the shaded regions overlap.
The points where boundaries meet are called the vertices (corners) of the region. To find them, solve the system of boundary equations.
Example 3: Graph the solution set of the system.
$\begin{cases}y\ge x^2\\y\le x+2\end{cases}$
Solution:
The first inequality is the region above the parabola ${y=x^2}$. The second is the region below the line ${y=x+2}$. Both boundaries are solid.
Find where the boundaries meet. Set ${x^2=x+2}$:
$\begin{align*}x^2-x-2&=0\\(x-2)(x+1)&=0\end{align*}$
So ${x=2}$ or ${x=-1}$. The vertices are ${(2,4)}$ and ${(-1,1)}$. The solution set is the region between the two curves.
Example 4: Graph the solution set of the system, and find its vertices.
$\begin{cases}x\ge 0,\quad y\ge 0\\x+2y\le 8\\3x+y\le 9\end{cases}$
Solution:
The first two inequalities keep the region in the first quadrant. Test ${(0,0)}$ in the other two: ${0\le 8}$ and ${0\le 9}$ are true. So the region is below both lines.
Three vertices are on the axes: ${(0,0)}$, the point ${(3,0)}$ where ${3x+y=9}$ meets the ${x}$-axis, and the point ${(0,4)}$ where ${x+2y=8}$ meets the ${y}$-axis.
The fourth vertex is where the two lines meet. From ${3x+y=9}$, ${y=9-3x}$. Substitute it:
$\begin{align*}x+2(9-3x)&=8\\-5x&=-10\\x&=2\end{align*}$
Then ${y=9-6=3}$. The vertex is ${(2,3)}$.
An application: linear programming
Businesses often want the largest profit, or the smallest cost, under some limits. When the limits are linear inequalities and the profit is a linear function, this is called linear programming. The best value always occurs at a vertex of the region:
The largest and smallest values of ${P=ax+by}$ on the solution set of a system of linear inequalities occur at vertices of the set.
Example 5: A small workshop makes tables and chairs. Each table takes ${1}$ hour of cutting and ${3}$ hours of finishing. Each chair takes ${2}$ hours of cutting and ${1}$ hour of finishing. Each day there are at most ${8}$ hours of cutting and ${9}$ hours of finishing. The profit is ${\$50}$ per table and ${\$40}$ per chair. How many of each should the workshop make per day for the largest profit?
Solution:
Let ${x}$ be the number of tables and ${y}$ the number of chairs. The limits are
- cutting: ${x+2y\le 8}$,
- finishing: ${3x+y\le 9}$,
- and ${x\ge 0}$, ${y\ge 0}$, since you cannot make a negative number.
This is the system of Example 4. The profit is ${P=50x+40y}$. Check each vertex:
| Corner | ${P=50x+40y}$ |
|---|---|
| ${(0,0)}$ | ${0}$ |
| ${(3,0)}$ | ${150}$ |
| ${(2,3)}$ | ${100+120=220}$ |
| ${(0,4)}$ | ${160}$ |
The largest profit is ${\$220}$ per day, from ${2}$ tables and ${3}$ chairs.
Summary
- To graph an inequality: draw the boundary (dashed for ${<}$ or ${>}$, solid for ${\le}$ or ${\ge}$), test a point, and shade the side that makes it true.
- The solution set of a system of inequalities is the overlap of all the shaded regions.
- The vertices of the region are found by solving the boundary equations together.
- In linear programming, the largest or smallest value of ${ax+by}$ occurs at a vertex.