Unit 1: FunctionsUnit 1: Functions1-1-22: Inequalities, Set-Builder Notation, : Inequalities, Set-Builder Notation,

and Iand Interval Notationnterval Notation

Recall: Solving Equations3x = 12 3 3x = 4

3x + 5 = 20 - 5 -5 3x = 15 3 3 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 > 12 3 3 x > 4

< or > mark the point w/ < or > mark the point w/

If x is on the left side, color in the direction of the arrow point on the inequality.

< means > 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 < -3 -3 -3 2x > -8_ x > -4 *the sign does not 2 2 change because the negative is not with the variable

Compound Inequalities2 types:1) And [3-headed monster, between]2) OR 8 < 3x + 5 < 20 -5 -5 -5 3 < 3x < 15 3 3 3 1 < x < 5

Included

Excluded

The graphs look like this:-3 < x < 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

x < -1 or x > 2-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Ex. 2x + 5 < 3 or 3x – 9 > 6

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

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Special Cases: Case #12x + 1 < 3 or 3x – 2 > -5 -1 -1 + 2 +2 2x < 2 3x > -3 x < 1 or x > -1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

These overlap to cover the entire line, so the solution set is all real numbers.

It can be written as

Special Cases: Case #22x + 1 < 3 and 3x +5 > 20 -1 -1 -5 -5 2x < 2 3x > 15 x < 1 and x > 5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

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 < 5 C. x < -6 D. 6 < x < 12

Interval notation:

Interval notation:

Interval notation: