Boole's inequality

Examples of Boole's inequality in the following topics:

• Elements of a Designed Study

  • Boole's inequality implies that if each test is performed to have type I error rate $\frac{\alpha}{n}$, the total error rate will not exceed $\alpha$.

• Nonlinear Systems of Inequalities

  • A system of inequalities consists of two or more inequalities, which are statements that one quantity is greater than or less than another.
  • A nonlinear inequality is an inequality that involves a nonlinear expression - a polynomial function of degree 2 or higher, as shown in Figure 1.
  • The most common way of solving one inequality with two variables x and y is to shade on a graph the region of points that satisfies the inequality.
  • Every inequality has a boundary line, which is the equation produced by changing the inequality relation to an equals sign.
  • If we have two inequalities, therefore, we shade in the "overlap" region, where both inequalities are simultaneously satisfied.

• Linear Inequalities

  • A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
  • For inequalities that contain variable expressions, you may be asked to solve the inequality for that variable.
  • A linear inequality looks like a linear equation, with the inequality sign replacing the equal sign.  
  • Step 2, since there is a variable on both sides of the inequality, choose to move the $-4x$, to combine the variables on the left hand side of the inequality, or move the $-6x$ to the right hand side of the inequality.
  • For the left inequality, subtract $3$ and then divide both sides by $-2$ (remember to reverse the inequality symbol: $-2x\leq-12$, $x\geq6$.

• How Income is Allocated

  • Recent growth in overall income inequality has been driven mostly by increasing inequality in wages and salaries.
  • Recent growth in overall income inequality has been driven mostly by increasing inequality in wages and salaries.
  • Globally, income inequality has increased over the last few decades.
  • Given that economic theory points to a decline in income inequality over time, the recent increase has led many researchers to conclude that we may be starting a new inequality cycle .
  • The Kuznets curve depicts the relationship between inequality and income; after hitting a market peak, inequality will decrease as income increases.

• The Significance of Social Inequality

  • Sociologists study many types of inequality, including economic inequality, racial/ethnic inequality, and gender inequality.
  • Sociology has a long history of studying stratification and teaching about various kinds of inequality, including economic inequality, racial/ethnic inequality, gender inequality, and other types of inequality.
  • Although inequality is everywhere, there are many controversies and questions about inequality that sociologists are interested in, such as where did inequality come from?
  • Do we justify inequality?
  • Can we eliminate inequality?

• Introduction

  • Sociology has a long history of studying stratification and teaching about various kinds of inequality, including economic inequality, racial/ethnic inequality, gender inequality, and other types of inequality.
  • Although inequality is everywhere, there are many controversies and questions about inequality that sociologists are interested in such as where did inequality come from?
  • Do we justify inequality?
  • Can we eliminate inequality?
  • We end with consequences of inequality and theories explaining global inequality.

• Compound Inequalities

  • Another type of inequality is the compound inequality, which can also be solved to find the possible values for a variable.
  • One type of inequality is the compound inequality.
  • A compound inequality is of the form:
  • It is customary (but not necessary) to write the inequality so that the inequality arrows point to the left:
  • Solve a compound inequality by balancing all three components of the inequality

• Graphs of Linear Inequalities

  • Graphing linear inequalities involves graphing the original line, and then shading in the area connected to the inequality.
  • Graphing an inequality is easy.
  • First, graph the inequality as if it were an equation.
  • Now if there is more than one inequality, start off by graphing them one at a time, just as was done with a single inequality.
  • To find solutions for the group of inequalities, observe where the area of all of the inequalities overlap.

• Rules for Solving Inequalities

  • Operations can be conducted on inequalities and used to solve inequalities for all possible values of a variable.
  • Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality.
  • To see these rules applied, consider the inequality $5 > -3$.
  • Solving an inequality gives all of the possible values that the variable can take to make the inequality true.
  • Recognize how operations on an inequality affect the sense of the inequality

• Solving Problems with Inequalities

  • Speculate on the number of solutions of a linear inequality.
  • (Hint: Consider the inequalities x < x−6 and x ≥ 9. )
  • If any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged.
  • If both sides of an inequality are multiplied or divided by the same positive number, the sense of the inequality remains unchanged.
  • If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed (change direction) in order for the resulting inequality to be equivalent to the original inequality.