Graphing Systems of
Linear Inequalities
Linear inequalities are linear
equations involving the inequality signs <,
>, ≤, ≥, and ≠.
Each equation divides the XY plane in half; the intersection of these
regions form the solution of a system of these equations.
Example: review
y_{1} = ½
x + 5
The region for y_{1} = ½
x + 5 is the line itself. Now let's change this equation into an
inequality:
y_{1} ≤ ½
x + 5
The graph is:
To determine the region specified by
the equation, choose any point x, draw a vertical line from x through
the line of the equation. Where this vertical line intersects the
line for the equation place a dot and label it with its corresponding
value for y.
In this case I chose x = 8, and drew
the line to y = 9. ( (8,9) is a solution to the equation.)
Now a point along this vertical line,
say below the intersection, has the same x value, but a smaller y
value. Since the equation requires those points to yield y values
less than or equal to 9, all these points are on the correct side of
that line and therefore satisfy the equation (dark black portion of
the vertical line).
A point above the intersection along
the (dashed gray) vertical line x = 8 will have a y value greater
than 9. All such points do not satisfy the equation and are therefore
not on the correct side of the line.
To indicate the region specified by the
equation we either draw arrows from the line of the equation in the
direction of the region (dark green short arrows) or we shade the
region with some color. The equal part of the inequality is shown
with a solid line for the equation. If only a less than or a greater
than relationship is given then a dashed line is used for the
equation of the line.
Another example:
y_{2} < −x + 3
Notice the dashed line used for the
equation, points on the line are not included because of the strict
less than relationship. At x = 9 a vertical line is drawn with the
intersection at y = 9. Since we require all y values less than this
value the region is below this line. Likewise moving up the vertical
line from the intersect gives y values greater than 9, which means
these values are outside the region. Notice that the bold short red
arrows start with circles on the line, again to indicate the points
on the line are excluded.
Another example:
The
line for y_{3} is sold indicating the equality. At x = 8 a
vertical line intersects at y = 2. The equation uses the relationship
greater than or equal to which indicates acceptable y values on the
vertical line must be greater than or equal to y = 2, and the line
for y_{3} must be solid. The region named by this equation
must therefore be above and including the line representing y_{3}
.
Three equations
are plotted next.
The
xy pairs in this region will satisfy all of these inequalities
simultaneously. If we graph all three regions and shade them, the
required solution will be where all three overlap. Doing this we
arrive at the following graph:
The solution is the region within the central
triangle, shown next in a
larger scale. You can see all three hatch patterns (vertical green,
horizontal red, and slanted blue) inside the triangle. This triangle
has two bold sides in green and blue and one dashed side in red. All xy pairs along the boundaries except the red boundary of this
triangle and those inside the triangle are solutions to this system
of linear inequalities.
Problems
involving linear constraints are solved by graphing these constraints
in this manner to determine the range of feasible solutions (the
intersection of the regions are these solutions.) A goal is usually
specified as a linear equation in terms of being maximized or
minimized and it's solution is one of the points of intersections of
the lines representing the constraints.
