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Rules for Solving Inequalities

The solution of a simple inequality is an interval consisting of all solutions satisfying the inequality and is represented as a part of the number line. While some of the rules for solving an inequality are same as those applied for solving equations, some rules are different. We need to learn specially these rules which differ from those applied for solving equations in order to get the correct solutions of inequalities.

Switching sides turns the inequality in to the opposite direction.
That is, a > b is equivalent to b < a.
We easily understand that 5 < 7  and 7 > 5 are equal.

Addition and subtraction properties of inequalities are similar to the corresponding properties of equations.

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Properties of Inequalities

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Addition Property of Inequalities:
For two real numbers a and b, a > b. Then,  a + c > b + c, where c is a real number.
In other words, adding same quantity to both the sides of an inequality results in an equivalent inequality. Let us look at an example where this property is used to solve an inequality.

Let us solve x - 5 > 3
Adding 5 to both the sides, we get
x > 8                             
This is an equivalent inequality which is the solution for the given inequality.

Subtraction Property of Inequality:
For real numbers a, b and c, if a > b, then a - c > b - c.
Subtracting same quantity from both the sides of an inequality gives an equivalent inequality.

Let us solve z + 3 < - 2
Subtracting 3 from both the sides, we get
z < -5
This is an equivalent inequality which is the solution.

On comparing two negative numbers, we see that the number with greater absolute value is indeed lesser. 
For example, 4 > 2. But, - 4 < - 2. This property of negative numbers influence the multiplication and division properties of inequalities.

Multiplication Property of Inequality:
For real numbers a, b and c, if a > b, then 
  1. ac > bc  if c > 0
  2. ac < bc  if c < 0

In other words, multiplication on either side by a negative quantity reverses the inequality symbol, while the symbol is retained when the inequality is multiplied on either side by a positive quantity.

Let us solve $\frac{1}{2}$ x > 5

Multiplying by 2 on both the sides, we get

2 ($\frac{1}{2}$ x) > $2 \times 5$

x > 10 
This is the solution for the inequality with the same symbol.

Division Property of Inequality:
Division property of inequalities is similar to the rules followed for multiplication.
For real numbers a, b and c and given that a > b, then

  1. $\frac{a}{c}$ > $\frac{b}{c}$   if c > 0
  2. $\frac{a}{c}$ < $\frac{a}{c}$   if c < 0

The division by a negative quantity on either side of the inequality reverses the inequality symbol as in the case of multiplication.

Let us solve 2x < 12
Dividing by 2 on either side, we get
2x $\div$ 2  < 12 $\div$ 2
x < 6
This is the solution to the inequality with the same inequality symbol.


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