- NOW WE'LL TAKE A LOOK AT TWO BASIC EXAMPLES
OF COMPOUND INEQUALITIES.
COMPOUND INEQUALITIES ARE TWO OR MORE INEQUALITIES
CONNECTED WITH THE WORDS "AND" OR "OR."
IN OUR FIRST EXAMPLE, NOTICE HOW THIS LOOKS LIKE TWO INEQUALITIES
SQUISHED TOGETHER.
THIS IS AN EXAMPLE OF TWO INEQUALITIES
CONNECTED BY "AND,"
SO IF WE START IN THE MIDDLE OF THIS INEQUALITY
WITH THE VARIABLE,
WE CAN READ THIS FROM LEFT TO RIGHT,O
AND WE CAN ALSO READ THIS FROM RIGHT TO LEFT.
SO IN BLUE, WE CAN SEE X IS LESS THAN OR EQUAL TO 7.
AND IN RED FROM RIGHT TO LEFT,
WE WOULD READ THIS X IS GREATER THAN -5.
AND BECAUSE OF THE WORD "AND,"
THE SOLUTION WILL BE ALL THE VALUES
THAT SATISFY BOTH OF THESE INEQUALITIES.
FOR EXAMPLE, IF YOUR BOSS SAYS YOU'LL GET A RAISE
IF YOU SHOW UP ON TIME EVERY DAY AND COMPLETE ALL OF YOUR WORK
THAT MEANS YOU'D HAVE TO COMPLETE BOTH TASKS
TO GET A RAISE.
SO WHAT WE'RE GOING TO DO IS GRAPH BOTH OF THESE INEQUALITIES
ON THE SAME NUMBER LINE
AND THEN DETERMINE GRAPHICALLY WHERE THEY'RE BOTH TRUE.
SO THE GRAPH X IS GREATER THAN -15,
WE'D HAVE AN OPEN POINT ON -5 AND AN ARROW TO THE RIGHT,
BECAUSE IT SAYS > -5.
AND X ALSO HAS TO BE LESS THAN OR EQUAL TO 7,
AND BECAUSE THIS INTERVAL DOES INCLUDE 7,
WE'D MAKE A CLOSED POINT ON 7.
AND THEN FOR NUMBER LESS THAN 7, WE'D HAVE AN ARROW TO THE LEFT.
NOW THIS IS NOT THE GRAPH OF OUR SOLUTION.
WE NEED TO INTERPRET THIS.
WE WANT TO DETERMINE WHICH INTERVAL SATISFIES
BOTH INEQUALITIES
OR WHICH NUMBERS HAVE BEEN GRAPHED TWICE,
BOTH IN BLUE AND IN RED.
AND YOU CAN SEE IT WOULD BE ALL THE NUMBERS FROM -5 TO +7,
NOT INCLUDING -5, BUT INCLUDING 7.
SO TO SHOW THIS, WE'LL ADJUST OUR GRAPH--
TO LOOK LIKE THIS.
AGAIN, IT DOES NOT INCLUDE +5,
BUT IT DOES INCLUDE +7
AND THEN ALL THE NUMBERS BETWEEN THESE TWO.
THEREFORE AN INTERVAL NOTATION
WOULD HAVE THE INTERVAL FROM -5 TO 7.
IT DOES NOT INCLUDE -5,
SO WE HAVE AN OPEN PARENTHESIS HERE, BUT IT DOES INCLUDE 7,
SO WE MAKE A SQUARE BRACKET HERE.
IN THE SECOND EXAMPLE,
IT'S A LOT MORE OBVIOUS THAT THIS IS AN "OR,"
BECAUSE IT'S WRITTEN AS AN "OR."
SO IF WE HAVE TWO INEQUALITIES CONNECTED BY "OR,"
THAT MEANS IF THE NUMBER IS IN EITHER INTERVAL,
IT SATISFIES THE COMPOUND INEQUALITY.
FOR EXAMPLE, IF YOUR BOSS SAYS YOU'LL GET A RAISE
IF YOU SHOW ON TIME EVERYDAY OR CLOSE A BIG ACCOUNT,
IF YOU DO EITHER OF THOSE, YOU'LL GET THE RAISE.
YOU DO NOT HAVE TO SATISFY BOTH.
SO ON THIS ONE, WE CAN GO STRAIGHT TO A GRAPH.
TO GRAPH X IS LESS THAN OR EQUAL TO -3,
THIS INTERVAL DOES INCLUDE -3,
SO WE'LL MAKE A CLOSED POINT ON -3,
ARROW TO THE LEFT,
OR X > 7, SO 7 IS NOT IN THIS INTERVAL,
SO WE'LL MAKE AN OPEN POINT ON 7.
AND THEN FOR > 7, WE MAKE AN ARROW TO THE RIGHT.
AGAIN BECAUSE THIS IS AN "OR,"
EITHER OF THESE INTERVALS WOULD SATISFY THE COMPOUND INEQUALITY.
SO THIS IS THE GRAPH OF OUR SOLUTION,
BUT NOW LET'S EXPRESS IT USING INTERVAL NOTATION.
AS WE MOVE LEFT ON THE NUMBER LINE,
WE'RE APPROACHING -INFINITY,
AND AS WE MOVE RIGHT ON THE NUMBER LINE,
WE'D BE APPROACHING +INFINITY.
AND THIS IS HELPFUL TO WRITE THE INTERVAL
USING INTERVAL NOTATION.
THE INTERVAL ON THE LEFT, WE WOULD HAVE -INFINITY
AND THEN -3 WITH A ROUNDED PARENTHESIS FOR -INFINITY,
BUT A SQUARE BRACKET FOR -3 BECAUSE IT INCLUDES -3,
OR IN BLUE, WE HAVE THE INTERVAL FROM 7 TO INFINITY,
NOT INCLUDING 7.
SO WE HAVE 7 TO INFINITY, ROUNDED PARENTHESIS HERE,
AS WELL AS NEXT TO THE 7 TO SHOW IT DOES NOT INCLUDE 7.
NOW I SHOULD MENTION THAT INSTEAD OF USING THE WORD "OR,"
SOMETIMES YOU'LL SEE TEXTBOOKS USE THE SYMBOL "U" FOR "UNION,"
AND IT DOES MEAN THE SAME THING, JUST SO YOU'RE AWARE.
I HOPE YOU FOUND THIS HELPFUL.