Inequalities

Things one must know:

• When you multiply or divide an inequality by a negative number, flip the inequality sign.
• You cannot multiply or divide an inequality by a variable, unless you know the sign of the number that the variable stands for.
  Set up two cases - the positive and the negative case - and solve each one separately to arrive at two possible scenarios.
• When taking reciprocals flip the sign unless the two sides have different signs. If you don't know the signs of both sides, you cannot take reciprocals. 
• You can only square inequalities when both sides have the same sign.
  If both sides are known to be positive, then do not flip the inequality sign when you square.
  If both sides are known to be negative, then flip the inequality sign when you square.
  If the sides are unclear or different, then you cannot square.
  
• Combining inequalities:
  1) Solve
  2) Simply so that all the inequality symbols point in the same direction
  3) Line up the common variables
  4) Combine by taking the more limiting upper and lower extremes
  
• You can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality.
• You can add inequalities together as long as the inequality signs are pointing in the same direction. You can multiply inequalities together only if both sides of inequalities are positive. You can never subtract or divide two inequalities. 
• Use extreme values for problems that involve multiple inequalities and the potential range of values for variables in the problem
• You should know how to solve problems involving both equations and inequalities by: (1) plugging the inequality into the equation using extreme values, and (2) plugging the equation into the inequality using standard algebra.
• Extreme value arithmetic operations (MGMAT 3rd book, p92)
  Flip the extreme value when subtracting, dividing and multiplying a negative number by an extreme value. Multiply two extreme values if you know that both are positive.
• Use extreme values to solve minimization and maximization problems - focus on the largest and smallest possible values for each of the variables. Set up a table with extreme values for all the variables. 
  Set terms with even exponents equal to zero when trying to solve optimization problems.
  
• When you see inequalities with zero on one side of the inequality, you should consider using positive/negative analysis to help solve the problem.
  Common inequality statements:
  xy > 0 implies that x and y are both positive or both negative
  xy < 0 implies that x and y have different signs
  x² - x < 0 or x² < x or 0 < x < 1
• Inequalities and absolute value
  | x + b | = c means the center point of the graph is -b and that x must be exactly c units away from -b.
  | x + b | < c means the center point of the graph is -b and that x must be less than c units away from -b.
  Solve algebraically:
  1) First scenario, simply remove the absolute value brackets and solve
  2) Second scenario, reverse the signs of the terms inside absolute value brackets, remove the brackets, and solve again.
  3) Combine the scenarios into one range of values
• Inequalities involving even exponents require you to consider two scenarios.
  √x² = |x| likewise consider that x can be negative or positive (i.e. positive x < 2 and negative x > -2
• If you have an inequality with zero on one side, see if you can factor the either side to create two factors that can be used for positive/negative analysis.
• Be wary of answer choices A or B on problems involving inequalities and variables. A seemingly in consequential statement, such as "x is positive" or "a is negative" if often crucial on inequality problems.