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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

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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

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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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