Graphing Systems of Linear Inequalities


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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

y1 = ½ x + 5





The region for y1 = ½ x + 5 is the line itself. Now let's change this equation into an inequality:


y1½ 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:

y2 < −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 y3 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 y3 must be solid. The region named by this equation must therefore be above and including the line representing y3 .




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.

 

 

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