INEQUALITY

INEQUALITY SYMBOLS

 

RULES OF INEQUALITIES

BOUNDED INTERVAL ON THE REAL NUMBER LINE

Let a and b be real numbers such that a < b. The following intervals on the real number line are bounded interval. The numbers a and b are the endpoints of each interval.

Notation	Interval Type	Inequality	Graph
[a ,b]	Closed
(a ,b)	Open
[a, b)	Half-open
(a, b]	Half-open

In closed interval, it contain both of its endpoints, a half open interval contains only one of its endpoints and an open interval neither of its endpoints. At times the solution to an inequality might result to an interval on the real line that is unbounded

 

 

UNBOUNDED INTERVAL ON THE REAL NUMBER LINE

Let a and b be real numbers. The following intervals on the real number line are unbounded intervals

Notation	Interval Type	Inequality	Graph
[a, ∞ )	Half –open		a[
(a, ∞)	Open		a (
(–∞, b]	Half open		] b
(–∞, b)	Open		)b
(–∞,∞)	Open

 

Example

Write an inequality to represent the interval and state whether the interval is bounded or unbounded.

1.                    [–2, 5]

2.                    (7, 15]

3.                    (9, ∞)

4.                    [–7, ∞]

Solution

SOLVING INEQUALITY

Linear inequalities are solved in the same way as linear equations. To solve the variable, we use properties of inequalities, which are similar to property of equality but there two important exceptions. When both sides of inequality are multiplied or divided by negative number, the direction of the inequality symbol must be reversed. 

Example 1

Solve the linear inequality

 Thus the solution consist of all real number that are greater than 9, the interval notation for this solution set is (9, ∞). The graph of this solution is shown below

                                                    

 

 

 

Example 2

Thus, the solution consist of real number that are less than or equal to . The interval notation for this solution set is (–∞,  ]. The graph of this solution set is shown below

 

 

 

 

Example 3

 Thus, the solution set consist of all real numbers that are less than or equal to 6. The interval notation for this solution is (– ∞,6]. The graph of this solution

 

 

 

 

 

DOUBLE INEQUALITY

Suppose we are given  we can simply write as . This form allows to combine two inequalities in one and also to solve the two give inequalities together

 

Example 4

Solve the inequality and sketch the graph the graph of its solution set

 

Solution

To solve the inequality, we can solve  separately and  and combine the result

 

 

 

INEQUALITY INVOLVING FRACTION

Example 5

Solution

 

	x < –5	–5 <x	< x < –1	x > -1
x + 1	–	–	–	+
x + 5	–	+	+	+
2x + 3	–	–	+	+
	–	+	–	+

Notice that  if and only if

 –5 < x <  or x > –1

Thus, the solution is given by

 

Example 6

Find the range of value of x for

Multiply both sides by 2(x + 2)²

 

 

	x < –2	–2 < x < 0	x > 0
x + 2	–	+	+
3x	–	–	+
3x (x + 2)	+	–	+

The required solution is x < –2 or x > 0

 

Example 7

Determine the range of value of x for which

 

Solution

	x < –4	–4 < x < 2	x > 2
x + 4	–	+	+
x – 2	–	–	+
(x + 4)(x – 2)	+	–	+

The required solution is x < –4 or x > 2

 

Example 8

Determine the range of value of x for which

 

Solution

	x< 2	2<x<3	3  < x < 7	x  > 7
x – 2	–	+	+	+
x – 3	–	–	+	+
x – 7	–	–	–	+
(x – 2)( x – 3)( x – 7)	–	+	–	+

The required solution is 2<x<3 or  x  > 7

 

MODULUS OR ABSOLUTE VALUES

The modulus of x is written as  can be defined

 

Example 9

 

Example 10

   

 

Example 11

Solve for x if

 

Solution

 

PROPERTIES OF ABSOLUTE VALUE

Example 12

Find x if

Solution

	x < –4	–4 < x < 3	x > 3
x – 3	–	–	+
x + 4	–	+	+
(x – 3)(x + 4)	+	–	+

The required solution is x < –4 or x > 3

 

 

Example 13

For what values of x is

Solution

 

 

	x < –7	–7 < x <– 5/3	x >–5/ 3
x +7	–	–	+
3x + 5	–	+	+
(x + 7)(3x + 5)	+	–	+

The required solution is –7 < x <– 5/3

 

INEQUALITY THEOREM

Given two real positive  number a and b ,  and  are also real numbers and

Example 14

Solution

i.  a and b are positive real number implies that  are real numbers then

 

 

 

Example 15

If x, y, z are positive numbers show that  

Solution

 

 

 

Example 16

If x, y, z are real numbers, show that

 

Solution

Example 17

If a, b are real and unequal, show that  Deduce that if a, b and c real and unequal then

 

Solution

 

 

 

Example 18

For any real number p, q, r, a, b, c prove

Solution

Example 19

Show that

Solution