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. This concept is crucial in understanding and solving various types of inequalities, including absolute value inequalities, rational inequalities, and quadratic inequalities.

5 Must Know Facts For Your Next Test

1. In a strict inequality, the inequality symbol used is either '<' (less than) or '>' (greater than), indicating that one value is strictly smaller or strictly larger than the other value.
2. Strict inequalities are often used to represent solutions that are not inclusive of the boundary values, unlike non-strict inequalities that include the boundary values.
3. Solving absolute value inequalities, rational inequalities, and quadratic inequalities often involves identifying the critical points or values where the inequality changes from true to false or vice versa.
4. The solutions to strict inequalities are typically represented as intervals on the number line, where the endpoints are not included in the solution set.
5. 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 certain constraints.

Review Questions

• Explain how strict inequalities are used in the context of solving absolute value inequalities.
  • When solving absolute value inequalities, strict inequalities are used to represent the solutions that are strictly greater than or strictly less than a certain value. For example, the inequality $|x - 3| < 5$ can be rewritten as the two strict inequalities: $x - 3 < 5$ and $x - 3 > -5$, which correspond to the solution set of $-2 < x < 8$. The strict inequalities ensure that the solutions do not include the boundary values, which are the values where the absolute value expression is equal to the constant.
• Describe how strict inequalities are used in the context of solving rational inequalities.
  • When solving rational inequalities, strict inequalities are used to identify the critical points or values where the inequality changes from true to false or vice versa. These critical points are typically the zeros of the denominator, which represent the values where the rational expression is undefined. The strict inequalities are then used to divide the number line into intervals, and the solution set is determined by evaluating the sign of the rational expression on each interval. The strict inequalities ensure that the solutions do not include the values where the rational expression is undefined.
• Analyze the role of strict inequalities in the context of solving quadratic inequalities.
  • In the process of solving quadratic inequalities, strict inequalities are used to represent the solutions that are strictly greater than or strictly less than the critical points, which are the values where the quadratic expression changes sign. These critical points are typically the solutions to the corresponding quadratic equation. The strict inequalities are then used to divide the number line into intervals, and the solution set is determined by evaluating the sign of the quadratic expression on each interval. The strict inequalities ensure that the solutions do not include the critical points, which are the boundary values where the inequality changes from true to false or vice versa.

Related terms

Inequality:

A mathematical statement that compares two expressions and indicates that one is greater than, less than, or not equal to the other.

Absolute Value Inequality:

An inequality that involves the absolute value of a variable or expression, which represents the distance of the value from zero on the number line.

Quadratic Inequality:

An inequality that involves a quadratic expression, which is a polynomial of degree two, and is often solved by using the sign of the discriminant or by graphing.