Unit 1: Functions 1-2: Inequalities, Set-Builder Notation, and Interval Notation

Recall: Solving Equations 3x = x = 4 3x + 5 = x = x = 5

More complicated equations used combining like terms or the distributive property.  Combining like terms: 1) must have the same variables 2) must have the same exponents on those variables  Distributive property a(b + c) = ab + ac * Used with combining like terms in many equations

Ex: 2x – 5(x + 2) = 8 - 2x

Ex: 12 – 5(2w - 3) = 3(2w – 5)

Ex: 8 + 5(3x – 4) = 7(x - 12)

Inequalities (basically the same) 3x > x > 4 mark the point w/ If x is on the left side, color in the direction of the arrow point on the inequality. means

Ex. 3x + 5 < 20 The answer is x < 5 Notice that the arrow point looks just like the inequality. <

Remember: ******  If you multiply or divide by a negative number, the inequality changes position. –3x < _9 x < x > -8_ x > -4 * the sign does not 2 2 change because the negative is not with the variable

Compound Inequalities 2 types: 1)And [3-headed monster, between] 2)OR 8 < 3x + 5 < < 3x < < x < 5 Included Excluded

The graphs look like this: -3 < x < x

Ex. 2x **the inequality sign doesn’t change– the negative is not with the “x”!!

Special Cases: Case #1 2x x -3 x These overlap to cover the entire line, so the solution set is all real numbers. It can be written as

Special Cases: Case #2 2x x 15 x These do not overlap at all. “AND” means that they must occur at the same time, but they never will, so the answer here is: “no solution. It can be written as

Set-Builder Notation  The inequalities:  Set-builder notation:  These are complete statements that say: 1) “the set of all x such that x is between negative 7 and 9” 2)“the set of all y such that y is between negative 8 and 2”

Interval Notation  Interval notation uses brackets: [ and ] to display included points  Interval notation uses parentheses: ( and ) to display excluded points.  Interval notation uses the infinity and negative infinity sign: and when the solution extends in a direction forever. Note: if you are using infinity, it always uses ( or ).

Examples using interval notation:

Ex. 1. Write the following inequalities in set-builder notation and in interval notation. A.x > 2 B. -3 < x < 5C. x < -6 D. 6 < x < 12

Interval notation: