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# Properties of Inequalities

An inequality is a statement about the relative size or order of two objects or about whether they are the same or not. It is just an approximate comparison of the two statements.

• The notation a < b means that a is less than b.
• The notation a > b means that a is greater than b.
• The notation a $\neq$ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.

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## Properties of Absolute Value Inequalities

Absolute values contain many important applications of inequalities. An absolute value represents the distance along the number line between the number and the 0.

The inequality, x < b, where b > 0 means that the distance between x and 0 is less than b.

Let x be an algebraic expression and b be a real number such that b $\geq$ 0.
• The solution of |x| < b are all values of x that lie between -b and b.
|x| < b iff x > -b and x < b.
• The solution of |x| > b are all values of x that are less than -b or greater than b.
|x| > b iff x < -b or x > b.

Above result is also valid for $\leq$ and $\geq$.

## Inequality Properties

Properties of inequalities are similar to the properties of equality. But, when each side of an equality is multiplied or divided by a negative number, the direction of the inequality must be reversed in order to maintain a true statement.

### Property 1: Addition Property of Inequality

For real numbers, a, b, c and d

a < b and c < d, then a + c < b + d

For example, if 5 < 10 and 3 < 7, then 5 + 3 < 10 + 7

8 < 17 (true statement)

### Property 2: Addition or subtraction of any constant.

If a < b, then a + c < b + c, a - c < b - cThe aptitude of an inequality is not altered when the similar number is added or subtracted from both faces of the inequality.

Consider 5x + 2 > 7x - 3

If we add 5 on both the sides, then it will be 5x + 7 > 7x + 2

So, here sign of inequality (>) is not changed.

### Property 3: Multiplication Property of Inequality

Multiply by a constant

• For any positive constant, c

a < b => ac < bc

• For any negative constant, -c

a < b => -ac > -bc

The aptitude of the inequality is not altered if both sides are multiplied or divided by the similar positive value.

Consider (2x + 3) > (x + 2)

If we multiply both the sides by 2, then it will be (4x + 6) > (2x + 4)

So, here sign of inequality (>) is not changed. The aptitude of the inequality is overturned if both sides are multiplied or divided by the similar negative value.

If a < b, then -a > -b

Consider 5 > -5.

If we multiply both the sides by -2, then it will be -10 < 10

So, here sign of inequality (<) is reversed.

### Property 4: Division Property of Inequality

1. Dividing each side of an inequality by a positive quantity produces an equivalent inequality.

For c > 0

If a < b, then $\frac{a}{c} < \frac{b}{c}$

2. Dividing each side of an inequality by a negative quantity produces an equivalent inequality in which inequality symbol is reversed.

For c < 0

If a < b, then $\frac{-a}{c} > \frac{-b}{c}$

Consider 2x > 4

If we divide both the sides by 2, then it will be x > 2

So, here sign of inequality (>) is not changed.

If we divide both the sides by -2, then it will be -x < -2

So, here sign of inequality (<) is reversed.

### Property 5: Transitive Property of Inequality

For real numbers a, b and c

a < b and b < c => a < c

For example, if 3 < 7 and 7 < 10, then 3 < 10.

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