Chapter 2.6 - Inequalities

Important Terms to Know:

• Inequality - A statement that two quantities or expressions are NOT equal
• Open Interval - an open interval is used to illustrate the solution of an inequality that consists of every real number between the two given restrictions, but NOT the given numbers surrounding x. Parentheses, "(" and ")", are used to denote an open interval.
• ex: 4<x<8 - (4, 8) 

• Closed Interval - a closed interval is used to illustrate the solution of an inequality that consists of every real number between the given restrictions, INCLUDING the given numbers surrounding x. Brackets, "[" and "]", are used to denote a closed interval. 

• ex: 3<x<7 - [3, 7] 

• Half-Open Interval - a half-open interval is used to illustrate the solution of an inequality containing both a less-then/greater-than and a less-than or equal-to/ greater-than or equal-to. 
• ex: 6>x>-3 - (6, -3] 

• Infinite Interval - an infinite interval is used to illustrate the infinite solutions of an inequality higher or lower than than the given restriction. parentheses are used next to the ∞ symbol. 
• ex: 5<∞ - (5, ∞] 

Properties of Inequalities

*It's important to remember that when multiplying or dividing both sides of an inequality by a negative real number REVERSES the inequality sign.

Solving an Inequality

ex #1

                 subtract 5 from both sides

                                   divide both sides by -6         

*  remember to reverse the inequality sign         

       x consists of all real numbers such that x>-2

                                                 write as interval 

ex #2

A real number x is a solution of the given inequality if and only if it is a solution of BOTH inequalities

Solve the first inequality: 

                              add 4 to both sides

                                                            simplify

                                      divide both sides by 2

                                                               simplify

                                          equivalent inequality

Now solve the second equation:

                                          add 4 to both sides

                                      divide both sides by 2

x is only a solution of the given inequality if and only if BOTH x>-1 AND x<3, or it could be written as -1<x<3

You could also solve the equation using this shorter, alternative method:

                                                    add 4 

                                                         simplify 

                                                     divide by 2

Solving a Rational Inequality

since the numerator is positive, the fraction is positive if and only if the denominator, x-2, is also positive. Thus, x-2>0 or, equivalently, x>2, and the solutions are all real numbers in the infinite interval (2, ∞). 

The Lens Formula 

If a convex lens has a focal length of f units and if an object is placed a distance p units from the lens with p>f, then the distance q from the lens to the image is related to p and f by the formula:

Union Symbol 

the union symbol unites two collections of points

Intersection Symbol

Properties of Absolute Values (b>0)

Good Luck everyone!! Rock on!!

By Austin Kellogg