HSSLIVE Plus One Maths Chapter 6: Linear Inequalities Notes

Linear inequalities are mathematical expressions that represent constraints and boundaries in various real-world scenarios. This chapter teaches students to solve and graph inequalities in one and two variables, work with systems of inequalities, and understand feasible regions. Through this study, students develop critical skills in modeling optimization problems, constraint analysis for resource allocation, and formulating scenarios where variables must satisfy specific conditions—skills particularly valuable in economics, business, and computer science.

Chapter 6: Linear Inequalities

Linear inequalities are mathematical expressions involving the inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Basic Properties:

1. If a < b, then:
   • a + c < b + c (adding same number to both sides)
   • a – c < b – c (subtracting same number from both sides)
   • a × c < b × c (if c > 0)
   • a × c > b × c (if c < 0)
   • a ÷ c < b ÷ c (if c > 0)
   • a ÷ c > b ÷ c (if c < 0)
2. Transitive Property: If a < b and b < c, then a < c.

Algebraic Solutions:

To solve a linear inequality like ax + b < c:

1. Isolate the variable term on one side.
2. Perform the necessary operations, remembering to flip the inequality sign when multiplying or dividing by a negative number.
3. Express the solution as an interval or graph it on a number line.

Graphical Representation:

• Solutions to linear inequalities in one variable are represented on a number line.
• Solutions to linear inequalities in two variables are represented as half-planes in the coordinate plane.
  • Ax + By < C represents the half-plane below the line Ax + By = C
  • Ax + By > C represents the half-plane above the line Ax + By = C

System of Linear Inequalities:

A system consists of multiple inequalities that must be satisfied simultaneously. The solution is the common region satisfying all inequalities.

Applications:

Linear inequalities are useful in:

• Linear programming problems
• Business optimization (maximizing profit, minimizing cost)
• Resource allocation problems
• Scheduling and planning