Graph solutions to advanced linear inequalities

Learning Objectives Solve multi-step linear inequalities that include variables on both sides. Solve linear inequalities that require use of the distributive property. Solve compound inequalities involving 'and' (intersections). Solve compound inequalities involving 'or' (unions). Graph the solution sets of multi-step and compound inequalities on a number line. Correctly use open and closed circles to represent the inclusion or exclusion of endpoints in a solution. Interpret and graph special cases, such as 'no solution' or 'all real numbers'. Ever tried to stay within a specific data limit on your phone plan or keep your game character's health above a critical level? 📱 You're using inequalities without even realizing it!...

TermDefinitionExample Compound InequalityTwo or more inequalities joined together by the words 'and' or 'or'.`x > 5` and `x ≤ 10` (which can be written as `5 < x ≤ 10`) is an 'and' inequality. `x < 0` or `x ≥ 3` is an 'or' inequality. Intersection ('and')The set of values that make *both* inequalities in a compound statement true. The graph is the overlapping region.The intersection of `x > -2` and `x < 3` is all the numbers between -2 and 3. Union ('or')The set of values that make *at least one* of the inequalities in a compound statement true. The graph combines all solutions from both inequalities.The union of `x ≤ -1` or `x > 4` includes all numbers less than or equal to -1, as well as all numbers greater th...

The Golden Rule of Inequalities (The 'Flip' Rule) If `a > b` and `c < 0`, then `ac < bc` and `a/c < b/c`. This is the most important rule. Whenever you multiply or divide both sides of an inequality by a NEGATIVE number, you MUST flip the direction of the inequality symbol (e.g., `>` becomes `<`). Solving 'And' Compound Inequalities `a < x + c < b` is solved by performing the same operation on all THREE parts. To isolate the variable in the middle of a three-part inequality, whatever you do to the middle, you must also do to the left and right sides. Solving 'Or' Compound Inequalities `ax + b < c` or `dx + e > f` Treat it as two separate problems. Solve the first inequality completely, then solve the second...