key term - Strict Inequality

Definition

A strict inequality is a mathematical relationship between two values where one value is strictly greater than or strictly less than the other value. It is denoted using the symbols '>' for 'greater than' and '<' for 'less than', and does not include the possibility of equality.

5 Must Know Facts For Your Next Test

1. Strict inequalities are used to model real-world situations where one quantity must be strictly greater than or strictly less than another quantity.
2. When solving applications with linear inequalities, strict inequalities are often used to represent constraints or limits on the variables.
3. The solution set for a strict inequality is the set of all values of the variable that satisfy the inequality, but does not include the boundary value where the inequality becomes an equality.
4. Graphically, the solution set for a strict inequality is represented by an open interval on the number line, as opposed to a closed interval for a non-strict inequality.
5. Strict inequalities are important in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints.

Review Questions

• Explain how strict inequalities differ from non-strict inequalities in the context of solving applications with linear inequalities.
  • The key difference between strict and non-strict inequalities is that strict inequalities do not include the boundary value where the inequality becomes an equality. For example, the strict inequality $x > 5$ represents all values of $x$ that are strictly greater than 5, whereas the non-strict inequality $x \geq 5$ includes the value $x = 5$ as part of the solution set. This distinction is important when solving real-world problems with linear inequalities, as the strict inequality may better model the constraints or limits on the variables.
• Describe the graphical representation of the solution set for a strict inequality and explain how it differs from the graphical representation of a non-strict inequality.
  • The solution set for a strict inequality is represented graphically as an open interval on the number line. This means that the boundary values where the inequality becomes an equality are not included in the solution set. For example, the strict inequality $x > 5$ would be represented by the open interval $(5, \infty)$, excluding the value $x = 5$. In contrast, the non-strict inequality $x \geq 5$ would be represented by the closed interval $[5, \infty)$, including the value $x = 5$. This graphical distinction is important when interpreting the solutions to linear inequality applications.
• Analyze the role of strict inequalities in optimization problems and explain how they can be used to find the maximum or minimum value of a function subject to certain constraints.
  • Strict inequalities play a crucial role in optimization problems, where the goal is to find the maximum or minimum value of a function subject to a set of constraints. In these problems, strict inequalities are often used to represent the constraints or limits on the variables. For example, in a problem where we want to maximize the profit of a product subject to a limited supply of raw materials, we might use strict inequalities to model the constraints on the amount of raw materials available. The solution set defined by these strict inequalities represents the feasible region, and the maximum or minimum value of the function within this feasible region is the optimal solution. Strict inequalities ensure that the boundary values are excluded from the feasible region, which is important for finding the true maximum or minimum value of the function.

Related terms

Inequality: A mathematical relationship where one value is either greater than, less than, greater than or equal to, or less than or equal to another value.

Linear Inequality:

An inequality that can be written in the form $ax + b > 0$ or $ax + b < 0$, where $a$ and $b$ are constants and $x$ is a variable.

Compound Inequality:

An inequality that combines two or more individual inequalities using the words 'and' or 'or'.