Linear Inequality

Definition

A linear inequality is a mathematical expression that represents an inequality between two linear expressions, such as $ax + b \geq cx + d$, where $a$, $b$, $c$, and $d$ are constants. These inequalities define a region in the coordinate plane that satisfies the given condition.

5 Must Know Facts For Your Next Test

1. Linear inequalities can be used to represent real-world situations, such as budget constraints, resource limitations, or other constraints on variables.
2. The solution set of a linear inequality is the set of all values of the variable that satisfy the inequality, and it is typically represented as a half-plane or a region in the coordinate plane.
3. Solving a linear inequality involves using the properties of inequalities, such as addition, subtraction, multiplication, and division, to isolate the variable and find the range of values that satisfy the inequality.
4. The graph of a linear inequality is a half-plane, and the boundary of the half-plane is the line that represents the equality version of the inequality.
5. Absolute value inequalities can be transformed into equivalent linear inequalities by considering the cases where the expression inside the absolute value is positive and negative.

Review Questions

• Explain how the solution set of a linear inequality is represented in the coordinate plane.
  • The solution set of a linear inequality is represented as a half-plane in the coordinate plane. The boundary of the half-plane is the line that represents the equality version of the inequality. The region of the half-plane that satisfies the inequality is the solution set. For example, the inequality $2x + 3 \geq 5$ has a solution set represented by the half-plane above the line $2x + 3 = 5$ in the coordinate plane.
• Describe the process of solving a linear inequality and how it differs from solving a linear equation.
  • Solving a linear inequality involves using the properties of inequalities, such as addition, subtraction, multiplication, and division, to isolate the variable and find the range of values that satisfy the inequality. This process is similar to solving a linear equation, but with some key differences. When solving a linear equation, the goal is to find a single value that satisfies the equation, whereas when solving a linear inequality, the goal is to find a range of values that satisfy the inequality. Additionally, the rules for manipulating inequalities, such as reversing the inequality sign when dividing or multiplying by a negative number, must be considered when solving linear inequalities.
• Explain how absolute value inequalities can be transformed into equivalent linear inequalities and the significance of this transformation.
  • Absolute value inequalities can be transformed into equivalent linear inequalities by considering the cases where the expression inside the absolute value is positive and negative. For example, the absolute value inequality $|x - 2| \leq 5$ can be rewritten as the system of linear inequalities $-5 \leq x - 2 \leq 5$, which can then be solved separately. This transformation is significant because it allows us to use the same techniques and properties used for solving linear inequalities to solve absolute value inequalities, which can be more complex. By breaking down the absolute value inequality into equivalent linear inequalities, we can find the complete solution set more easily.

Related terms

Inequality:

An inequality is a mathematical statement that compares two expressions using symbols such as $\geq$, $\leq$, $>$, or $<$.

Linear Expression: A linear expression is a polynomial expression of the first degree, consisting of a constant and a variable term, such as $2x + 3$.

Absolute Value Inequality:

An absolute value inequality is a type of inequality that involves the absolute value of a variable, such as $|x - 2| \leq 5$, which defines a region around a specific value.