Solving for a system of linear inequalities requires finding values for
each of the variables such that all equations are satisfied.
LEARNING OBJECTIVES [ edit ]
Solve systems of linear inequalities by graphing
Solve systems of linear inequalities using non-graphical methods
KEY POINTS [ edit ]
To solve a system graphically, draw and shade in each of theinequalities on the graph, and then
look for an area in which all of the inequalities overlap, this area is the solution.
If there is no area in which all of the inequalities overlap, then the system has no solution.
To solve a system non-graphically, find the intersection points, and then find out relative to those
points which values still hold for the inequality. Narrow down these values until mutually
exclusive ranges (no solutions) are found, or not, in which the solution is within your final range.
TERMS [ edit ]
mutually exclusive
Describing multiple events or states of being such that the occurrence of any one implies the nonoccurrence of all the others.
subset
With respect to another set, a set such that each of its elements is also an element of the other set.
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Graphical Method
To solve a system oflinearinequalities, if possible, the easiest way to do this is by graphing .
However, graphing is only possible if there are two or three variables. For two variables,
when graphing, first draw all of the lines of
the inequalities as if they were
an equation, drawing a dotted line if it is
either < or > , and a solid line if it is either ≤ or ≥ . After the line has been drawn,
shade in, or indicate with hash marks, the
area that corresponds to the inequality.
For instance, if it is < or ≤ , shade in the
area below the line. If it is > or ≥ shade in
the area above the line.
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Interactive Graph: System of Linear Equations
Graph of system of linear equations (red) and (blue). In a system of linear equalities, the solution is at the
point of intersection. In comparison, the solution to a system of linear inequalities is defined by the space
according to the inequalities.
Once all of the inequalities have been drawn and shaded in, to find solutions to the system
one needs to find areas in which all of the inequalities overlap each other. For example, given
the system:
⎧y
⎪
⎨y
⎪
⎩y
≥
≥
≤
−2x − 1
2x + 1
x + 2
Draw each of the lines and shade in, or indicate, their corresponding inequalities, and then
look to see what parts overlap. As can be seen in , the shaded part in the middle is where all
three inequalities overlap.
Interactive Graph: Solution to Linear Inequality System
Graph of the three inequalities $y\ge ­2x­1$ (red), $y\ge 2x+1$ (blue), and $y\le x+2$ (purple). This is a
graph showing the solutions to a linear inequality system. Note that it is the overlapping areas of all
three linear inequalities.
If all of the inequalities of a system fail to overlap over the same area, then there is no
solution to that system. For instance, given the following system:
⎧
⎪
1
⎧y
⎪
⎨y
⎪
⎩y
≥
≥
≤
1
2x + 2
x + 1
Again, draw all the inequalities and shade in, or indicate somehow, the area that the
inequality covers. Notice that in this graph , there is no part of the graph where all three
inequalities overlap. There are plenty of areas where two of the three overlap at a time, but
that is not enough, all three must overlap for those points to be a solution to the system.
Interactive Graph: Linear Inequality System With No Solutions
Graph of the three inequalities $y\ge 1$ (red), $y\ge 2x+2$ (blue), and $y\le x+1$ (purple). This is a
graph showing a system of linear inequalities that has no solution as there is no point in which the areas
of all three inequalities overlap. Contrast this with the graph "Solution to Linear Inequality System".
Non-Graphical Method
Sometimes one may not wish to graph the equations, or simply cannot due to the number of
variables. In this situation, findintervals in which certain variables satisfy the system by
looking at two equations at a time. First, find the intersection point(s) of two of the equations,
if there is no intersection, then the two inequalities are either mutually exclusive, or one of
the inequalities is a subset of the other. For a simple example, x > 2 and x < 1 are mutually
exclusive, whereas x > 2 and x > 1 has x > 2 as a subset of x > 1. If they are mutually exclusive,
then there is no solution.
Once an intersection point is found, determine on which side(s) of the intersection point the
inequalities hold. For example, if there are two equations y ≥ −2x and y ≤
intersection is found to occur at x
−1
=
3
x + 1
, the
.
Look at any point greater than that x value (say 0) to see that the first equation gives y ≥ 0 ,
and the second gives y ≤ 1 . Since these two equations are not mutually exclusive, these two
equations are satisfied for any x ≥
−1
3
.
If the entire system had a third equation as well: y ≤ 4 , then next find the intersection points
between this new equation and the other two. The intersection between y ≤ 4 and y
≥ −2x occurs at x = −2 , and that it is satisfied when x ≥ −2 . Note that the x ≥
−1
3
found
earlier is more restrictive.
Therefore, ignore the new inequality for x. Now the last set y ≤ 4 and y ≤
x = 3
x + 1
intersects at , further, by looking at x=2 and x=4, it is seen that for these two inequalities, x ≤ 3 .
Now for the system to hold true, −1
3
≤ x ≤ 3 , and to find valid y values, simply choose any x
value in that range, and plug them in to get the range on the y values.
The non-graphical method is much more complicated, and is perhaps much harder to
visualize all the possible solutions for a system of inequalities. However, having too many
equations or too many variables, may be the only feasible method.