key term - Linear Inequality

Definition

A linear inequality is a mathematical expression that represents an inequality relationship, such as less than or greater than, between a linear expression and a constant value. Linear inequalities are used to model and solve real-world problems involving constraints and boundaries.

5 Must Know Facts For Your Next Test

1. Linear inequalities can be solved using the same techniques as solving linear equations, such as adding, subtracting, multiplying, or dividing both sides by a constant.
2. The solution set of a linear inequality is the set of all values of the variable that make the inequality true.
3. The graph of a linear inequality is a half-plane, with the boundary line representing the equality case.
4. Systems of linear inequalities can be used to model and solve real-world problems with multiple constraints, such as resource allocation or optimization problems.
5. The intersection of the solution sets of multiple linear inequalities represents the feasible region, which is the set of all points that satisfy all the inequalities simultaneously.

Review Questions

• Explain how to solve a linear inequality, and describe the steps involved.
  • To solve a linear inequality, you can use the same steps as solving a linear equation, such as adding, subtracting, multiplying, or dividing both sides by a constant. The goal is to isolate the variable on one side of the inequality sign. The solution set is the set of all values of the variable that make the inequality true. For example, to solve the inequality $2x + 5 \geq 15$, you would subtract 5 from both sides to get $2x \geq 10$, and then divide both sides by 2 to get $x \geq 5$. The solution set is all real numbers greater than or equal to 5.
• Describe the process of graphing a linear inequality and explain the significance of the boundary line.
  • To graph a linear inequality, you first need to determine the boundary line, which represents the equality case. This boundary line is graphed as a solid or dashed line, depending on whether the inequality is strict (< or >) or inclusive (≤ or ≥). The solution set of the inequality is represented by the half-plane on one side of the boundary line. The choice of which half-plane to shade depends on the direction of the inequality symbol. For example, for the inequality $3x - 2y \leq 6$, the boundary line is $3x - 2y = 6$, and the shaded region represents all points $(x, y)$ that satisfy the inequality.
• Analyze the role of linear inequalities in solving real-world applications and describe how systems of linear inequalities can be used to model and solve optimization problems.
  • Linear inequalities are widely used to model and solve real-world problems that involve constraints or boundaries. For example, a business may need to determine the optimal production levels of two products given limited resources, such as raw materials, labor, and budget. This can be represented as a system of linear inequalities, where each inequality represents a constraint. The feasible region, which is the intersection of the solution sets of all the inequalities, represents the set of all possible production levels that satisfy the constraints. By maximizing or minimizing an objective function, such as profit, within the feasible region, the business can determine the optimal solution to the problem. Systems of linear inequalities are powerful tools for modeling and solving optimization problems in various fields, including economics, logistics, and resource allocation.

Related terms

Inequality: A mathematical relationship between two expressions or values that are not equal, such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

Linear Expression: An algebraic expression that is a sum of constant terms and variables with degree one, such as 2x + 3y - 5.

Graphing Inequalities: The process of representing the solution set of a linear inequality on a coordinate plane, typically using a half-plane or a shaded region.