Linear Inequalities

Learning Objectives

• Understand and solve linear inequalities in one variable using algebraic manipulation and represent solutions graphically on a number line.
• Graphically solve linear inequalities in two variables using half-planes, boundary lines, and test points.
• Comprehend the properties of inequalities, including the effects of addition, subtraction, and multiplication/division by positive or negative numbers on the inequality sign.
• Graphically represent solutions of inequalities on a number line, distinguishing between strict and slack inequalities using open and closed circles.
• Solve systems of inequalities and represent their solutions graphically, focusing on the intersection of solution sets.
• Apply inequalities to solve real-world problems, such as determining minimum or maximum values under given constraints.
• Solve double inequalities and express solutions in interval notation, such as solving $-8 \leq 5x - 3 < 7$.
• Identify the feasible region for two-variable inequalities, including boundary line inclusion/exclusion, for inequalities like $ax + by \leq c$ or $ax + by \geq c$.

Chapter Notes

Linear Inequalities in One Variable

• Definition: An inequality involving a single variable, expressed in the form $ax + b < 0$, $ax + b > 0$, $ax + b \leq 0$, or $ax + b \geq 0$.
• Example: Solve $30x < 200$.
  • Solution:
    • Divide both sides by 30: $x < \frac{200}{30}$
    • Simplify: $x < \frac{20}{3}$
    • If $x$ is a natural number, solution set: {1, 2, 3, 4, 5, 6}
    • If $x$ is an integer, solution set: {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

Linear Inequalities in Two Variables

• Definition: An inequality involving two variables, expressed in the form $ax + by < c$, $ax + by > c$, $ax + by \leq c$, or $ax + by \geq c$.
• Graphical Solution:
  • Represented using half-planes and boundary lines.
  • Use test points to determine which half-plane satisfies the inequality.

Properties of Inequalities

• Addition/Subtraction: Equal numbers can be added to or subtracted from both sides without changing the inequality.
• Multiplication/Division:
  • By a positive number: Inequality sign remains the same.
  • By a negative number: Inequality sign is reversed.

Graphical Representation of Inequalities

• On a Number Line:
  • Strict Inequality ($<$ or $>$): Use open circles.
  • Slack Inequality ($\leq$ or $\geq$): Use closed circles.

System of Inequalities

• Definition: A set of two or more inequalities with the same variables.
• Solution: The intersection of the solution sets of the individual inequalities.
• Graphical Representation: Use overlapping regions on a graph to find the common solution area.

Applications of Inequalities

• Real-World Problems: Use inequalities to determine minimum or maximum values under given constraints.
• Example: A student needs a minimum of 70 marks in the annual exam to achieve an average of at least 60 marks.

Double Inequalities and Interval Solutions

• Example: Solve $-8 \leq 5x - 3 < 7$.
  • Solution:
    • Split into two inequalities: $-8 \leq 5x - 3$ and $5x - 3 < 7$.
    • Solve each separately and combine the solutions.
    • Solution set: $-1 \leq x < 2$.

Feasible Region for Two-Variable Inequalities

• Definition: The region that satisfies all given inequalities in a system.
• Boundary Line Inclusion/Exclusion:
  • Solid Line: Includes the boundary (inequality with $\leq$ or $\geq$).
  • Dashed Line: Excludes the boundary (inequality with $<$ or $>$).

Examples

• Example 1: Solve $4x + 3 < 6x + 7$.
  • Solution:
    • Rearrange: $-2x < 4$
    • Divide by -2: $x > -2$
    • Solution set: $(-2, \infty)$
• Example 2: Solve $3x - 2 \geq x + 1$.
  • Solution:
    • Rearrange: $2x \geq 3$
    • Divide by 2: $x \geq \frac{3}{2}$
    • Solution set: $[\frac{3}{2}, \infty)$