What is the Rule for Inequalities?

The rule for inequalities, similar to equations, involves maintaining balance, but with a focus on preserving the direction of the inequality.

Understanding Inequality Rules

Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to show the relationship between quantities that are not necessarily equal. Here's a breakdown of the core rules:

Operations that Preserve the Inequality

These operations, when applied to both sides, will leave the inequality symbol unchanged:

• Addition: Adding the same quantity to both sides of an inequality maintains the inequality.
  • Example: If a < b, then a + c < b + c.
• Subtraction: Subtracting the same quantity from both sides of an inequality keeps the inequality true.
  • Example: If a > b, then a - c > b - c.
• Multiplication by a Positive Number: Multiplying both sides of an inequality by a positive number does not change the inequality.
  • Example: If a ≤ b and c > 0, then ac ≤ bc.
• Division by a Positive Number: Dividing both sides of an inequality by a positive number also leaves the inequality unchanged.
  • Example: If a ≥ b and c > 0, then a/c ≥ b/c.

Operations that Reverse the Inequality

These operations require flipping the inequality symbol:

• Multiplication by a Negative Number: Multiplying both sides of an inequality by a negative number reverses the direction of the inequality symbol.
  • Example: If a < b and c < 0, then ac > bc.
• Division by a Negative Number: Dividing both sides of an inequality by a negative number also reverses the direction of the inequality symbol.
  • Example: If a > b and c < 0, then a/c < b/c.

Key Takeaways

Operation	Effect on Inequality Symbol	Example
Adding a quantity	No change	If x < 5, then x + 2 < 7
Subtracting a quantity	No change	If x > 5, then x - 2 > 3
Multiplying by a positive number	No change	If x ≤ 3, then 2x ≤ 6
Dividing by a positive number	No change	If x ≥ 4, then x/2 ≥ 2
Multiplying by a negative number	Reverse the symbol	If x < 3, then -x > -3
Dividing by a negative number	Reverse the symbol	If x > 6, then -x/2 < -3

Important Note from the reference: Adding or subtracting the same quantity from both sides leaves the inequality symbol unchanged. Multiplying or dividing by a positive number on both sides leaves the inequality symbol unchanged.

These rules are essential for solving and manipulating inequalities in algebra and beyond. Always remember to pay close attention to whether you are multiplying or dividing by a negative number!