Mastering Linear Inequalities

Equivalent Inequalities

• Inequalities that have the same solution set are called equivalent inequalities.
• The techniques used to create equivalent inequalities include adding or subtracting the same quantity from both sides or multiplying or dividing both sides by a positive number.
• Creating equivalent inequalities helps to maintain the relationship between two expressions or variables while solving equations or inequalities.

Addition Property of Inequality

• The addition property states that if you add the same number to both sides of an inequality, the inequality remains true.
• For example, if a < b, then a + c < b + c for any real number c.
• This property is fundamental in solving inequalities, allowing for the isolation of a variable term.

Multiplication Property of Inequality

• The multiplication property indicates that multiplying both sides of an inequality by a positive number preserves the inequality's direction.
• If a < b and c > 0, then ac < bc.
• However, if both sides are multiplied by a negative number, the direction of the inequality reverses, meaning if a < b and c < 0, then ac > bc.
• Understanding this property is crucial for correctly manipulating inequalities in algebraic expressions.