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Solving Inequalities with Two Variables

 A linear inequality in two variables contain two variables and any one of the inequality symbols > , < , $\geq$ or $\leq$.
2x - 3y < 5, 4x + y $\geq$ -7 and y $\leq$ 3x - 4 are some of the examples of inequalities in two variables. An ordered pair (x, y) which satisfies an inequality of two variables is a solution of the inequality.

For example, let us consider the inequality in two variables x - 2y $\leq$ 8.

Let us check whether the ordered pair (4, 1) is a solution of the inequality.
Substituting x = 4 and y = 1 in the inequality we get,
4 - 2 (1) $\leq$ 8 => 4 - 2 $\leq$ 8  or 2 $\leq$ 8 which is a true statement.
Hence the ordered pair (4, 1) is a solution of the given inequality. 
The complete solution set of an inequality is a half plane region consisting of all ordered pairs satisfying the inequality on x - y Plane.
The linear equation corresponding to the inequality represents the line that divides the x - y plane into two half planes and serves as a boundary line for the region which represents inequality.

Steps for graphing a linear inequality in two variables:

  1. Graph the boundary line of the Inequality. The boundary line is marked with broken line if the inequality symbol does not include '=' sign, that is when the inequality has only symbols < or > and not $\leq$ or $\geq$.
  2. Choose a test point from the plane and check whether it is a solution of the inequality.
  3. Shade the appropriate half plane.

Graph the inequality x + y < 2.
The boundary line for the inequality is x + y = 2  or y = - x + 2.
Let us choose the test point as the origin (0, 0).
Substituting in the inequality we get,
0 + 0 < 2  which is a true statement.
Hence (0, 0) is a solution of the inequality and the graph should contain the point (0, 0).
The half plane containing (0, 0) is shaded and the graph of the inequality is shown below.

Solving Inequalities with Two Variables

Boundary line is shown broken indicating the points on the line are not solutions of inequality. A point on the boundary line should not be taken as the test point as it may not be a solution and will not indicate the region that is to be shaded to represent the graph.

Inequality in two variables representing real life situation:
Suppose you are making Motifs using silk fabric. You want to make some Flower Motifs and some Animal Motifs. You have at present 120 sq.ft of silk material. 

For each Flower Motif you require, 2 sq.ft of material and each Animal Motif requires 3 sq.ft of material. 

Write an inequality to represent the possible of number of Motifs in each type that can be made in the available material.

Let us assume that x number of Flower Motifs and y numbers of Animal Motifs are made out of the required material.

Then, the material required for making x number of Flower Motifs = 2x sq.ft
Material required for making y number of Flower Motifs = 3y sq.ft.
Hence x and y should satisfy the inequality 2x + 3y $\leq$ 120,  with restrictions x $\geq$ 0 and y $\geq$ 0.

Related Calculators
Graphing Inequalities in Two Variables Calculator Linear Equations in Two Variables Calculator
Solving Two Step Equations Calculator for Solving Inequalities

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