A compound inequality is a statement that combines two or more simple inequalities using the logical connectives 'and' or 'or'. These inequalities are used to describe a range of values that satisfy multiple conditions simultaneously.
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Compound inequalities can be used to solve linear inequalities and applications involving linear inequalities.
The solution set of a compound inequality consisting of 'and' conditions is the intersection of the solution sets of the individual inequalities.
The solution set of a compound inequality consisting of 'or' conditions is the union of the solution sets of the individual inequalities.
Graphing is a useful technique for visualizing and solving compound inequalities, as it helps identify the overlapping or combined solution sets.
Compound inequalities can be used to model real-world scenarios, such as setting price ranges, determining feasible solutions, or describing the limits of a system.
Review Questions
Explain the difference between a simple inequality and a compound inequality, and provide an example of each.
A simple inequality is a single comparison between a variable or expression and a constant using an inequality symbol, such as $x > 5$ or $y \leq 3$. In contrast, a compound inequality combines two or more simple inequalities using logical connectives like 'and' or 'or'. For example, a compound inequality could be $-2 \leq x \leq 4$, which represents the range of values for $x$ that are greater than or equal to -2 and less than or equal to 4.
Describe the process of solving a compound inequality that uses the 'and' connective, and explain how the solution set is determined.
To solve a compound inequality that uses the 'and' connective, you need to solve each individual inequality separately and then find the intersection of the 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 in this case is the interval $-3 \leq x \leq 5$.
Explain how compound inequalities can be used to model real-world scenarios, and provide an example of such an application.
Compound inequalities can be used to model various real-world situations that involve multiple constraints or conditions. For instance, a company might use a compound inequality to determine the acceptable price range for a product, such as $20 \leq p \leq 30$, where $p$ represents the price. This would ensure that the price is not too low to be profitable, nor too high to be unaffordable for customers. Compound inequalities can also be used to describe the limits or feasible region of a system, such as the acceptable range of temperatures and humidity levels for the proper functioning of a piece of equipment.
Related terms
Simple Inequality: A simple inequality is a mathematical statement that compares a variable or expression to a constant using one of the inequality symbols: <, >, ≤, or ≥.
Logical Connectives: Logical connectives, such as 'and' and 'or', are used to combine two or more simple inequalities into a compound inequality.