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Understanding Inequalities in Mathematics

Introduction to Inequalities

• The instructor introduces the topic of inequalities, explaining that they are similar to equations but do not include an equal sign.

• Key inequality symbols discussed include greater than, less than, greater than or equal to, and less than or equal to.

Properties of Inequalities

First Property: Addition

• When adding a value to both sides of an inequality, the direction of the inequality remains unchanged. Example: If x < 11 , then x + 6 < 17 .

Second Property: Subtraction

• Similar to addition, subtracting a value from both sides keeps the inequality's direction intact. For instance, if x + 7 > 3 , it simplifies correctly without changing the symbol.

Third Property: Division by Positive Numbers

• Dividing both sides of an inequality by a positive number does not alter its direction. Example given is dividing by 3 leading to x geq 3 .

Fourth Property: Multiplication by Positive Numbers

• Multiplying both sides by a positive number also preserves the direction of the inequality. An example shows that multiplying results in x < 12 .

Fifth Property: Division by Negative Numbers

• Dividing by a negative number reverses the direction of the inequality. This is illustrated with -14 / -7 = 2 , resulting in x > -2 .

Sixth Property: Multiplication by Negative Numbers

• Finally, multiplying both sides of an inequality by a negative number also requires reversing the symbol's direction. The example concludes with x < -3 .