Linear Inequalities

Recall that a function of the form f ( x ) = ax + b , a, b ∈ R are constants, is called a linear function, because its graph is a straight line. Here a is the slope of the line and b is the y -intercept. If a = , then x -intercept x = − b a is obtained by solving f ( x ) = ax + b = . But there are situations where we need to consider linear inequalities.

For example to describe a statement like “ A tower is not taller than fifty feet. ” If x denotes the height of the tower in feet, then the above statement can be expressed as x ≤ . Basic Algebra Example . Our monthly electricity bill contains a basic charge, that is independent of units consumed and a charge that depends on the units consumed.

Let us say Electricity Board charges Rs. as basic charge and charges Rs. for each unit we use. If a person wants to keep his electricity bill below Rs.

, then what should be his electricity usage? Let x denote the number of units used. Note that x ≥ . Then, his electricity bill is Rs.

+ x . The person wants his bill to be below Rs. . Let us solve the inequality + x < .

Thus, x < ; which gives ≤ x < . The person should keep his usage below units in order to keep his bill below Rs. . Example .

Solve x − ≤ x + for x . We have x − ≤ x + ; which is equivalent to x ≤ . Hence we have x ≤ ; the solution set is ( −∞ , ] . We can also solve the above inequality graphically.

Let us consider the graphs of f ( x ) = x − and g ( x ) = x + (See Figure . ). Now, find all the x -values for which the graph of f is below the graph of g . – – – – – – – – – g ( x ) = x + f ( x ) = x – Figure .

Example . Solve the following system of linear inequalities. x − ≥ , x − ≤ . Note that x − ≥ implies x ≥ , by multiplying both sides by / we get x ≥ .

Similarly, x − ≤ implies x ≤ and hence x ≤ . So the solution set of x − ≥ , x − ≤ is the intersection of [ , ∞ ) and ( −∞ , ] . Clearly, the intersection of these intervals give [ , ] .