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# Inequalities

**Inequalities**- Use the symbol " < " or " > " to express the relationship of less than or large than is called inequalities.

**Example**

**Less Than**- Any numbers less than one can be expressed as x < 1.

**Example**

**Large Than**- Any numbers large than zero can be expressed as x > 0

**Example**

**Less Than or Equal To**- Any numbers that are less than one or equal to one can be expressed as x < 1 or x = 1.

**Example**

**Large Than or Equal To**- Any numbers that are large than zero or equal to zero can be expressed as x > 0 or x = 0.

**Example**

**Large Than p and Less Than q**- Any numbers that are large than p and less than q can be expressed as x > p and x < q.

**Example**- If p = -1 and q = 2, then x > -1 and x < 2.

**Large Than or Equal To p and Less Than or Equal To q**- Any numbers that are large than or equal to p and less than or equal to q can be expressed as x >= p and x <= q.

**Example**- If p = -1 and q = 2, then x >= -1 and x <= 2.

**Addtion Property for Inequalities**- If a < b, then a + c < b + c, in which a, b and c are real numbers.

**Example**

**Subtraction Property for Inequalities**- If a < b, then a - c < b - c, in which a, b and c are real numbers.

**Example**

**Property of Multiplying of a Positive Number for Inequalities**- If a < b, then a c < b c, in which a, b and c are real numbers and c is a positive number.

**Example**

**Property of Multiplying a Negative Number for Inequalities**- If a < b, then a c > b c, in which a, b and c are real numbers and c is a negative number.

**Example**

**Property of Dividing a Positive Number for Inequalities**- If a < b, then a / c < b / c, in which a, b and c are real numbers and c is a positive number.

**Example**

**Property of Dividing a Negative Number for Inequalities**- If a < b, then a / c > b / c, in which a, b and c are real numbers and c is a negative number.

**Example**

**One Variable First Degree Inequalities**- The inequalities in which there is only one variable with degree of one is called one variable first degree inequalities.

**Example**

**Solution of One Variable First Degree Inequalities**- If ax + b < 0, then x < -b/a (a is not zero). if ax + b > 0, then x > -b/a (a is not zero).

**Example****Steps to Solve Inequalities**- (1). Removing parenthesis.
- (2). Moving all veriable terms to the left side of the inequality and moving all numerical terms to the right side of the inequality.
- (3). Combining similar terms.
- (4). Dividing by the coefficient of the variable on both side.
- (5). If multiply -1 on both side, then the inequality will change its direction.