≥ − . Show the graph of the solutions on number line. Solution We have ≥ or ≥ or ( x – ) ≥ ( x – ) LINEAR INEQUALITIES or x – ≥ x – or x ≥ or x ≥ The graphical representation of solutions is given in Fig . .
Fig . Example The marks obtained by a student of Class XI in first and second terminal examination are and , respectively. Find the minimum marks he should get in the annual examination to have an average of at least marks. Solution Let x be the marks obtained by student in the annual examination.
Then ≥ or + x ≥ or x ≥ Thus, the student must obtain a minimum of marks to get an average of at least marks. Example Find all pairs of consecutive odd natural numbers, both of which are larger than , such that their sum is less than . Solution Let x be the smaller of the two consecutive odd natural number, so that the other one is x + . Then, we should have x > ...
( ) and x + ( x + ) < ... ( ) Solving ( ), we get x + < i.e., x < ... ( ) From ( ) and ( ), we get < x < Since x is an odd number, x can take the values , , , and . So, the required possible pairs will be ( , ), ( , ), ( , ), ( , ) MATHEMATICS EXERCISE .
. Solve x < , when x is a natural number. x is an integer. .
Solve – x > , when x is a natural number. x is an integer. . Solve x – < , when x is an integer.
x is a real number. . Solve x + > , when x is an integer. x is a