Linear Inequalities

The notion of being equal is one of the very basic things in mathematics. The next obvious thing, which comes to our minds is its negation, what about inequality?

Generally speaking, we encounter equalities, when we solve equations, where one or more unknowns are present. For example, {3x + 9 = 7} gives {x = \frac {-2}{3}}. This is known as the solution of the equation.

By replacing the {equality} sign with either of the inequality signs, {> \ (greater \ than)} or {< \ (less than)}, we get an inequality.

When the original equation is a linear equation, on replacement, we get a linear inequality. For example, {3x + 9 > 7} is a linear inequality, whose solution is {x > \frac {-2}{3}}. Thus, any real number greater than {\frac {-2}{3}} satisfies this inequality.

The set of all values of {x}, which satisfy the inequality is known as the solution set of the inequality. It can be a finite set or an infinite set, depending on the situation.

  • Properties

I) If {b<c}, then {b \pm a < c \pm a}.

II) If {b < c} and {a>0}, then {ba < ca} and {\frac b a < \frac c a}.

III) If {b < c} and {a < 0}, then {ba > ca} and {\frac b a > \frac c a}

  • Graphical Representation of Linear Inequality in Single Variable

Linear inequalities of the form {ax + b > c} can be represented on the real number line. Their solution set is of the form {\frac {c-b}{a}}.

  • Linear Inequalities in 2 Variables

When 2 unknowns are present in an inequality, the solution set consists of pairs. For example, consider the inequation

{2x + 3 y \ge 5}

This inequality contains 2 variables {x} and {y}. To get the solution set graphically, we first draw the line {2x+3y = 5}.

Note that any line not passing through origin divides the {XOY} plane into 2 regions, one of which contains the origin. To get the solution set, we substitute the coordinates of origin in the inequality and check, whether they satisfy the inequality. If they do, the origin side of the line is the solution set. If they don’t, then the non-origin side of the plane is the solution set.

If the inequation contains the signs {\ge} or {\le} instead of {>} or {<}, then the line representing the inequality also forms a part of the solution set.

  • Feasible Region and Solution Set

When incidences/ phenomena in the real life are to be modeled using linear inequalities,  one has to consider more than 1 inequlities. By studying various possibilities out of the solution set, one has to arrive at the optimum solution.

Since linear equations are geometrically represented by straight lines, more than one linear inequality form a region. The region, whose points satisfy all the linear inequalities is known as the feasible region.

These regions can be closed sets (convex polygons) or open sets (unbounded convex sets). More details will be known to you in further discussions.

  • A Convex Set

A set of points in a plane is said to be convex, when the line segment joining any 2 points of the set lies in the set. These sets may be bounded (convex polygons) or unbounded.

  • WHY to Study Linear Inequations? (Linear Programming/ Linear Optimization)

A linear inequality can be used to represent different variables affecting the outcome of a business. The objective of the business being maximizing the profit / minimize the cost, one forms an equation with the linear inequalities as the constraints. The optimum value of the solution is then chosen as the best solution and later business activities can be planned accordingly.

Next : Linear Programming Problems

Posted in XI

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