key term - Linear Inequality

Definition

A linear inequality is a mathematical statement that represents an inequality between two linear expressions. It is a type of inequality that involves variables and coefficients in a linear relationship, where the variables are raised to the first power.

5 Must Know Facts For Your Next Test

1. Linear inequalities can have one or more variables and can be represented in the form $ax + by \geq c$, where $a$, $b$, and $c$ are constants.
2. The solution set of a linear inequality can be represented graphically as a half-plane on the coordinate plane.
3. When solving linear inequalities, the same operations can be performed on both sides of the inequality without changing the direction of the inequality symbol.
4. Graphing linear inequalities in two variables involves graphing the corresponding linear equation and then shading the appropriate half-plane based on the inequality symbol.
5. The solution set of a system of linear inequalities is the intersection of the individual solution sets of each inequality.

Review Questions

• Explain the process of solving a single linear inequality with one variable.
  • To solve a single linear inequality with one variable, you would first isolate the variable by performing inverse operations on both sides of the inequality, maintaining the direction of the inequality symbol. This may involve adding, subtracting, multiplying, or dividing both sides by a constant. Once the variable is isolated, the solution set can be expressed as a range of values that satisfy the inequality.
• Describe the steps involved in graphing a linear inequality in two variables.
  • To graph a linear inequality in two variables, you would first graph the corresponding linear equation by plotting points or using the slope-intercept form. Then, based on the inequality symbol (≤, <, ≥, >), you would shade the appropriate half-plane on the coordinate plane. The resulting shaded region represents the solution set of the linear inequality.
• Analyze the relationship between the solution sets of a system of linear inequalities and the intersection of their individual solution sets.
  • The solution set of a system of linear inequalities is the intersection of the individual solution sets of each inequality. This means that the values that satisfy all the inequalities in the system are the only values that belong to the final solution set. The graphical representation of a system of linear inequalities is the overlapping region of the individual half-planes, and the points in this overlapping region constitute the solution set.

Related terms

Inequality:

An inequality is a mathematical statement that compares two values or expressions, using symbols such as <, >, ≤, or ≥.

Linear Expression: A linear expression is a polynomial expression with variables raised to the first power, such as 3x + 5 or -2y + 7.

Solution Set:

The solution set of a linear inequality is the set of all values of the variable(s) that make the inequality true.