Understanding Inequality: Types and Causes

• Inequality refers to disparity in resource, opportunity, or outcome distribution within a population.
• Mathematical inequality compares two expressions that are not necessarily equal.

Types of Inequality

• Income inequality involves the unequal distribution of income among individuals or households.
• Wealth inequality involves the unequal distribution of assets (property, stocks) minus liabilities.
• Social inequality involves differences in access to social goods and services like education, healthcare, and justice.
• Health inequality involves disparities in health outcomes and access to healthcare.
• Gender inequality involves unequal treatment or opportunities based on gender.
• Racial inequality involves unequal treatment or opportunities based on race.

Causes of Inequality

• Market forces are supply and demand factors influencing wages and prices.
• Technological change involves automation and digital technologies increasing demand for skilled labor while displacing low-skilled workers.
• Globalization involves increased trade and capital flows benefiting some while harming others through job displacement and wage stagnation.
• Education: Unequal access to quality education perpetuates inequality.
• Policy and institutions: Tax policies, regulations, and institutional structures can exacerbate or mitigate inequality.
• Discrimination: Bias based on race, gender, or other characteristics can limit opportunities.
• Inheritance: Wealth can be passed down through generations, concentrating resources.

Measuring Inequality

• The Gini coefficient measures income distribution, ranging from 0 (perfect equality) to 1 (perfect inequality).
• The Palma ratio is the ratio of the richest 10% of the population's income to the poorest 40%.
• Income shares refer to the proportion of total income held by different population segments (e.g., top 1%, bottom 50%).

Consequences of Inequality

• Economic consequences include reduced economic growth, decreased consumer demand, and increased financial instability.
• Social consequences include increased crime rates, decreased social cohesion, and political instability.
• Health consequences include increased stress, reduced access to healthcare, and poorer health outcomes.
• Political consequences include increased political polarization, decreased trust in government, and weakened democratic institutions.

Addressing Inequality

• Progressive taxation involves taxing higher incomes at a higher rate to redistribute wealth.
• Minimum wage laws set a minimum wage to ensure low-skilled workers receive a living wage.
• Education and training involves increasing access to quality education and training to improve skills and opportunities.
• Social safety nets provide social insurance programs like unemployment benefits and welfare to protect vulnerable populations.
• Anti-discrimination policies enact laws and policies to prevent discrimination based on race, gender, or other characteristics.
• Strengthening labor unions empowers workers to negotiate for better wages and working conditions.
• Corporate governance reforms promote responsible corporate behavior and reduce excessive executive compensation.

Mathematical Inequality: Basics

• Inequality is a relation that holds between two values when they are different.
• Notation includes:
  • a ≠ b means that a is not equal to b
  • a < b means that a is less than b
  • a > b means that a is greater than b
  • a ≤ b means that a is less than or equal to b
  • a ≥ b means that a is greater than or equal to b

Solving Inequalities

• Solving inequalities involves finding the range of values that satisfy the inequality.
• This is similar to solving equations, but with some key differences.
• When multiplying or dividing by a negative number, the inequality sign must be reversed.
• If -2x < 6, then x > -3

Linear Inequalities

• A linear inequality involves a linear expression.
• For example: 3x + 2 < 5x - 1
• Solving linear inequalities involves isolating the variable on one side of the inequality.

Compound Inequalities

• Compound inequalities involve two or more inequalities combined.
• These can be connected by "and" or "or."
• "And" means both inequalities must be true simultaneously.
• "Or" means at least one of the inequalities must be true.
• A compound inequality example : 2 < x ≤ 5

Absolute Value Inequalities

• Absolute value inequalities involve absolute value expressions.
• |x| < a means -a < x < a.
• |x| > a means x < -a or x > a

Quadratic Inequalities

• Quadratic inequalities involve quadratic expressions.
• Example: x² - 3x + 2 > 0.
• Solving quadratic inequalities involves finding the roots of the quadratic equation and testing intervals.

Graphing Inequalities

• Inequalities can be graphed on a number line or in the coordinate plane.
• On a number line, a closed circle indicates that the endpoint is included, while an open circle indicates that the endpoint is not included.
• In the coordinate plane, a dashed line indicates that the boundary is not included, while a solid line indicates that the boundary is included.

Systems of Inequalities

• A system of inequalities involves two or more inequalities that must be satisfied simultaneously.
• The solution to a system of inequalities is the region where all inequalities are true.
• This region is often referred to as the feasible region.

Properties of Inequalities

• Addition: If a < b, then a + c < b + c
• Subtraction: If a < b, then a - c < b - c
• Multiplication by a positive number: If a < b and c > 0, then ac < bc
• Multiplication by a negative number: If a < b and c < 0, then ac > bc (inequality sign reverses)
• Division by a positive number: If a < b and c > 0, then a/c < b/c
• Division by a negative number: If a < b and c < 0, then a/c > b/c (inequality sign reverses)

Interval Notation

• Interval notation represents a set of numbers.
• (a, b) represents all numbers between a and b, excluding a and b.
• [a, b] represents all numbers between a and b, including a and b.
• (a, ∞) represents all numbers greater than a.
• [a, ∞) represents all numbers greater than or equal to a.
• (-∞, b) represents all numbers less than b.
• (-∞, b] represents all numbers less than or equal to b.