1. Properties of Inequalities
The expression
"a < b" is read as "a is less than b"
while the expression
"a > b" is read as "a is greater than b"
These signs define what is known as the sense of the inequality (indicated by the direction of the sign).
Two inequalities are said to have
(a) the same sense if the signs of inequality point in the same direction; and
(b) the opposite sense if the signs of inequality point in the opposite direction.
Examples
The inequalities x + 3 > 2 and x + 1 > 0 have the same sense.
So do the inequalities 3x - 1 < 4 and x2- 1 < 3.
The inequalities
x - 4 < 0 and x > - 4
have the opposite sense and so do the inequalities
2x + 4 > 1 and 3x 2 - 7 < 1.
The solution of an inequality consists of all the values of the variable that make the inequality a true statement.
Conditional inequalities are those which are true for some, but not all, values of the variable.
Absolute inequalities are those which are true for all values of the variable.
A solution of an inequality consists of only real numbers as the terms "less than or greater than" are not defined for complex numbers.
Examples
The inequality x + 1 > 0 is true for all values of x greater than -1.
Hence the solution of the inequality is written as x > -1 and so this is a conditional inequality.
However, the inequality x2 + 1 > 0 is true for all values of x and hence is an absolute inequality.
Graphical Representation of Inequalities
Examples
(a) To show x > 2 graphically, we use an open circle at 2 on the number line and a line to the right of this point, with an arrow pointing to the right:
The open circle shows that the point is not part of the indicated solution.
(b) To show x ≤1 graphically, we use a solid circle at 1 on the number line and a line to the left of this point, with an arrow pointing to the left:
The solid circle shows that the point is part of the indicated solution.
Properties of Inequalities
Property 1 - Adding or Subtracting a Number
The sense of an inequality is not changed when the same number is added or subtracted from both sides of the inequality.
Example
Using the inequality:
9 > 6
adding 4 to both sides gives
9 + 4 > 6 + 4
i.e. 13 > 10 which is still true
subtracting 12 from each side gives
9 - 12 > 6 - 12
i.e. -3 > -6 which is still true
Property 2 - Multiplying by a Positive Number
The sense of the inequality is not changed if both sides are multiplied or divided by the same positive number.
Example
Using the inequality:
8 < 15
Multiplying both sides by 2 gives
8 × 2 < 15 × 2
i.e. 16 < 30 which is still true
Dividing both sides by 2 gives
8/2 < 15/2
i.e. 4 < 7.5 which is still true
Property 3 - Multiplying by a Negative Number
The sense of the inequality is reversed if both sides are multiplied or divided by the same negative number.
Example
We start with the inequality 4 > -2.
Multiplying both sides by -3 gives
4 × -3 > -2 × -3
-12 > 6 which is not true
Hence the correct solution should be
4 > -2
4 × -3 < -2 × -3
-12 < 6 (Note the change in the sign used)
Similarly dividing both sides by -2 gives
4 > -2
4 ÷ -2 < -2 ÷ -2
-2 < 1 (Note the change in the sign used)
Property 4 - n-th Power
If both sides of an inequality are positive and n is a positive integer, then the inequality formed by the n-th power or n-th root of both sides have the same sense as the given inequality.
Example
Using the inequality:
9 > 6
Squaring both sides gives
92 > 62
i.e. 81 > 36 which is still true
Taking square root of each side gives
i.e. 3 > 2.45 which is still true
[Note: √9 does not equal ±3. By convention, we take the positive square root only. See the discussion at √16 - how many answers?]
Exercise
Graph the given inequality on the number line:
1 < x ≤ 4
Solution:
We need to have an open circle for 1, since it is not included, but a closed circle for 4, since it is included.
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