Linear Inequalities A linear inequality is an inequality that involves one or more linear expressions. These expressions can be combined using addition,...
A linear inequality is an inequality that involves one or more linear expressions. These expressions can be combined using addition, subtraction, multiplication, and division, but they cannot contain any variables in the exponent or root functions.
Examples:
x + 3 > 10
2x - 4 ≥ 8
3x / 4 ≤ 6
Properties of linear inequalities:
Adding or subtracting the same amount from both sides of an inequality preserves its direction.
Multiplying or dividing both sides by the same non-zero constant results in an inequality with the same direction.
The solution set to a linear inequality is the set of all values of x that satisfy the inequality.
Solving linear inequalities:
We can solve linear inequalities by isolating the variable on one side of the inequality sign using algebraic manipulations.
We can then apply the properties of linear inequalities to simplify the inequality and determine the solution set.
Applications of linear inequalities:
Linear inequalities have wide applications in various fields, including:
Physics: describing motion, equilibrium, and forces
Economics: modeling supply and demand, predicting market behavior
Finance: calculating investment risks and portfolio diversification
Real-world scenarios: scheduling, resource allocation, and quality control
Additional Notes:
Linear inequalities can be solved graphically by plotting the inequalities on a coordinate plane.
The solution set to an inequality is always a subset of the domain of the linear expressions involved.
Solving linear inequalities requires strong algebraic skills, including inequalities, inequalities, and properties of linear expressions