Linear Inequality

Definition

A linear inequality is a mathematical expression that shows the relationship between two values where one value is not equal to the other, typically expressed in the form of $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$. These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.

5 Must Know Facts For Your Next Test

1. Linear inequalities can be graphed on a number line (for one variable) or in a coordinate plane (for two variables), showing all possible solutions that satisfy the inequality.
2. When graphing a linear inequality, if the inequality is strict (using < or >), the boundary line is dashed; if it is inclusive (using \leq or \geq), the boundary line is solid.
3. In a system of linear inequalities, the solution set is determined by the intersection of all individual inequalities, forming the feasible region.
4. Linear inequalities can arise in real-life applications, such as determining budget limits, resource allocations, and constraints in optimization problems.
5. When solving linear inequalities, it’s important to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

Review Questions

• How do you determine the solution set for a linear inequality in one variable?
  • To determine the solution set for a linear inequality in one variable, you first isolate the variable on one side of the inequality. Then, you can express the solution as an interval on a number line. For example, for an inequality like $2x - 3 < 5$, you would add 3 to both sides and then divide by 2 to find that $x < 4$. The solution set would then be represented as $(-\infty, 4)$ on the number line.
• What is the significance of a dashed versus a solid boundary line when graphing linear inequalities?
  • The distinction between dashed and solid boundary lines is crucial when graphing linear inequalities. A dashed line indicates that points on the line are not included in the solution set for strict inequalities (like < or >), while a solid line indicates that points on the line are included for inclusive inequalities (like \leq or \geq). This visual representation helps clarify which solutions satisfy the inequality.
• Evaluate how linear inequalities can be utilized in real-world situations such as budgeting or resource allocation.
  • Linear inequalities play a vital role in real-world applications like budgeting and resource allocation by allowing individuals and organizations to set constraints based on their resources and needs. For example, if a company has a budget limit represented by an inequality such as $x + 2y \leq 100$ (where x and y represent quantities of different products), it can help determine how many units of each product can be produced without exceeding budgetary constraints. By analyzing these inequalities graphically through their feasible regions, decision-makers can visualize potential outcomes and make informed choices about allocating limited resources effectively.

Related terms

Boundary Line: The line that represents the equality part of a linear inequality when graphed, which divides the coordinate plane into two regions.

Feasible Region: The area on a graph where all constraints of a system of inequalities are satisfied, representing all possible solutions.

Objective Function: A function that needs to be maximized or minimized in linear programming, usually subject to various constraints represented by linear inequalities.