Method

Solving a System of Linear Inequalities Graphically

A system of linear inequalities can be solved by graphing all inequalities on the same coordinate plane and then finding the region of intersection, if any. For example, consider the following system.

{x + y < 7 x + 2 y \leq 10 (I) (II)

To solve the system graphically, these three steps can be followed.

Write the Inequalities in Slope-Intercept Form

To simplify the process of graphing the inequalities, start by writing them in slope-intercept form.

{x + y < 7 x + 2 y \leq 10 (I) (II)

SubIneq

$(I):$ $LHS - x < RHS - x$

{y < - x + 7 x + 2 y \leq 10

SubIneq

$(II):$ $LHS - x \leq RHS - x$

{y < - x + 7 2 y \leq - x + 10

DivIneq

$(II):$ $LHS / 2 \leq RHS / 2$

{y < - x + 7 y \leq - 0.5 x + 5

Graph the Inequalities

To graph the inequalities, begin with writing the corresponding boundary lines. The boundary line of the first inequality is

y = - x + 7 .

Inequality I : Boundary Line I : y < - x + 7 y = - x + 7

Since the inequality is strict, the boundary line should be dashed and, in this case, the shaded region is the one below the line.

Drawing y=-x+7 dashed and the region below it is shaded

Similarly, the boundary line of the second inequality is

y = - 0.5 x + 5 .

Inequality II : Boundary Line II : y < - 0.5 x + 5 y = - 0.5 x + 5

Since the inequality is non-strict, its boundary line is solid. In addition, the region to be shaded is the one below the line. This inequality will be graphed on the same coordinate plane.

Drawing y=-0.5x+5 solid and the region below it is shaded

Find the Overlapping Region

Notice that there is a region where the solution sets of the inequalities overlap. All the points in this region satisfy both inequalities simultaneously. Therefore, the overlapping region is the solution set of the system. In the next graph, only the common region is shaded.

Since the boundary lines in their entirety are not part of the solution set, they can be cropped to show only the edges of the overlapping region, or the exceeding parts can be drawn with lower opacity.

Keep in mind that if there is no overlapping region, the system has no solution.

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