Linear inequalities and interval notation

• Solve linear inequalities in one variable
  • Remember when to reverse the inequality (mult or divide by same negative number)
• Test solutions (above and below range)
• Write solutions in range notation
• Graph solutions on number line

• A linear inequality is an inequality relation applied to two linear expressions. There are four types of linear inequalities corresponding to the four types of inequality relations: x < y, x > y, x ≼ y, and x ≽ y.

  We say that a number is a solution for the given inequality in variable x if substituting that number for x makes the inequality true. The collection of all such numbers forms the solution set for the inequality.

(Interval Notation). The interval notation (a, b) describes the set of all real numbers between a and b. When we want to include the endpoints, we use square brackets: [a, b]. When we want the interval to extend indefinitely, we use the infinity sign: (a, ∞).

Inequalities are equivalent if they have the same solution sets.

THEOREM (Addition Property for Inequalities). Adding the same number on both sides of an inequality produces an equivalent inequality.

THEOREM (Multiplication Property for Inequalities). Multiplying both sides of an inequality by the same positive number produces an equivalent inequality. Multiplying both sides of an inequality by the same negative number and then reversing the inequality sign produces an equivalent inequality.

Formally, the following three inequalities in x are all equivalent to each other for all real numbers a and all positive numbers m: x < a, mx < ma, −mx > −ma

Similarly, the following three inequalities in x are all equivalent to each other: x ≤ a, mx ≤ ma, −mx ≥ −ma