Compound Inequality

Definition

A compound inequality is a statement that involves two or more simple inequalities combined using the logical connectives 'and' or 'or'. It represents a range of values that satisfy all the individual inequalities within the compound statement.

5 Must Know Facts For Your Next Test

1. Compound inequalities can be solved using the same principles as solving simple inequalities, but with additional steps to handle the logical connectives.
2. When solving a compound inequality with 'and', the solution set is the intersection of the individual inequality solutions.
3. When solving a compound inequality with 'or', the solution set is the union of the individual inequality solutions.
4. Absolute value inequalities can be rewritten as compound inequalities to solve them.
5. Quadratic inequalities can also be solved using the same compound inequality techniques, with additional steps to handle the quadratic expression.

Review Questions

• Explain the difference between a simple inequality and a compound inequality, and provide an example of each.
  • A simple inequality is a statement that compares a variable or expression to a constant using one of the inequality symbols (<, >, ≤, ≥). For example, $x > 5$ is a simple inequality. A compound inequality is a statement that involves two or more simple inequalities combined using the logical connectives 'and' or 'or'. For example, $-2 \leq x \leq 4$ is a compound inequality that represents the range of values where $x$ is greater than or equal to $-2$ and less than or equal to $4$.
• Describe the process of solving a compound inequality with the 'and' connective, and explain how the solution set is determined.
  • To solve a compound inequality with the 'and' connective, you need to solve each simple inequality separately and then find the intersection of the individual solution sets. For example, to solve the compound inequality $-3 \leq x \leq 5$, you would first solve the inequality $x \geq -3$ and then solve the inequality $x \leq 5$. The solution set would be the values of $x$ that satisfy both inequalities, which is the interval $[-3, 5]$ represented in interval notation.
• Analyze how the solution process for a compound inequality with the 'or' connective differs from the 'and' connective, and provide an example to illustrate the difference.
  • The key difference in solving a compound inequality with the 'or' connective is that the solution set is the union of the individual inequality solutions, rather than the intersection. For example, to solve the compound inequality $x < -2 \text{ or } x > 3$, you would first solve the inequality $x < -2$ and then solve the inequality $x > 3$. The solution set would be the values of $x$ that satisfy either of the individual inequalities, which is the interval $(-\infty, -2) \cup (3, \infty)$ represented in interval notation. This is in contrast to the 'and' connective, where the solution set is the intersection of the individual inequality solutions.

Related terms

Simple Inequality: A simple inequality is a statement that compares a variable or expression to a constant using one of the inequality symbols (<, >, ≤, ≥).

Logical Connectives: Logical connectives, such as 'and' and 'or', are used to combine two or more simple inequalities into a compound inequality statement.

Interval Notation: Interval notation is a way to represent the range of values that satisfy a compound inequality using set notation, such as [a, b] or (a, b).