Basic Concept of Inequalities

What are Inequalities? The symbols ‘>’ and ‘<’ are used to compare two numbers. For example, (a) ‘7 is greater than 4’ can be written as (b) ‘2 is less than 5’ can be written as ‘7 > 4’. ‘2 < 5’. Both ‘>’ and ‘<’ are inequality signs.

This is known as the law of trichotomy. Besides ‘>’ and ‘<’, there are other three common inequality signs: ‘>’, ‘<’ and ‘=’.

When two expressions are connected with an inequality sign, an inequality is formed. All the examples in the previous table are inequalities. Note: For any two numbers x and y, the meanings of the following pairs of inequalities are the same.

Use an inequality to represent the statement in each of the following. (a) 12 is less than x. (b) 3 + y is not greater than 40. (c) The weight of a box (w kg) is not less than 5 kg. (a) 12 x < (b) 40 3 + y £ (c) 5 w ³

Follow-up question 10. than less not is times 3 (b) 2. greater (a) . statements following the of each for inequality an up Set y x 2 x > 10 3 y ³ (c) 7 is less than or equal to b minus 5. 5 7 - b £

Solutions of Inequalities and their Graphical Representations For an inequality in one unknown, the values of the unknown that satisfy the inequality are called the solutions of the inequality. Let us consider the inequality x > 0. When x = 1, the inequality becomes 1 > 0, which is true. ∴ x = 1 is a solution of the inequality.

In fact, there is an infinite number of solutions to the inequality x > 0. –2 –1 1 2 3 4 5 Consider the above number line. All the numbers to the right of 0 on the number line are greater than 0. Therefore, they are the solutions of the inequality x > 0 (i.e. solutions are 1, 2, 3, 4, 5, etc.). Actually, we can represent the solutions of inequalities graphically.

o For example, of Solutions 1. > x 0. of solutions are 0, excluding of Solutions 1. > x 0. of solutions are 0, excluding right the to values All > x included not is ’ ‘ that means circle hollow The (ii) 0. of right the to numbers all are solutions indicates pointing arrow (i) : Note o > x in the solution.

· of Solutions 2. ³ x 0. of solutions are 0, including right the to of Solutions 2. ³ x 0. of solutions are 0, including right the to values All ³ x included is ’ ‘ that means circle solid The : Note · in the solution.

of Solutions 3. < x 0. of solutions are 0, excluding left the to values All < x 0. of left the to numbers all are solutions that indicates pointing arrow The : Note < x

of Solutions 4. £ x 0. of solutions are 0, including left the to values All £ x

Follow-up question 1. Determine whether x = –2 is a solution of the inequality 2x + 3 –5 . Solution When x = –2, L.H.S. = 2(–2) + 3 = –1 R.H.S. = –5 The inequality is satisfied. ∵ –1 –5 is true. x = –2 is a solution of the inequality 2x + 3 –5 . ∴

Follow-up question (cont’d) 2. Represent the solutions of the following inequalities graphically.