- WELCOME TO A LESSON IN WHICH WE'RE GIVEN AN INTERVAL
USING AN INEQUALITY AND THEN ASKED TO GRAPH
THE INTERVAL AS WELL AS EXPRESS THE INTERVAL
USING INTERVAL NOTATION.
SO FOR OUR FIRST EXAMPLE, THE INEQUALITY
IS "X" IS GREATER THAN +2.
NOTICE HOW IT DOES NOT SAY "X" IS EQUAL TO +2,
JUST GREATER THAN, AND THEREFORE
WE HAVE AN OPEN POINT ON +2 TO INDICATE THAT 2 IS NOT
IN THE INTERVAL.
AND THEN WE HAVE AN ARROW TO THE RIGHT WHERE VALUES
GET GREATER THAN +2.
AS WE MOVE TO THE RIGHT, WE WOULD APPROACH
POSITIVE INFINITY.
AND IT'S HELPFUL TO INCLUDE THIS BECAUSE IT HELPS US
RIGHT THE INTERVAL USING INTERVAL NOTATION.
USING INTERVAL NOTATION, THE INTERVAL
IS FROM TWO TO INFINITY.
IT DOES NOT INCLUDE +2, SO WE SAY THE INTERVAL IS OPEN ON +2.
AND WE USE A ROUNDED PARENTHESIS INSTEAD OF A SQUARE BRACKET.
AND THEN FOR INFINITY AND NEGATIVE INFINITY,
WE ALWAYS USE A ROUNDED PARENTHESIS.
IT'S ALSO IMPORTANT TO RECOGNIZE
THAT THE ORDER OF THESE IS IMPORTANT.
THE TWO HAS TO COME FIRST, AND THE INFINITY
HAS TO COME SECOND IN THE SAME ORDER
IT WOULD OCCUR ON THE NUMBER LINE.
NOW, LOOKING AT THE SECOND EXAMPLE,
NOTICE HOW WE HAVE "X" IS LESS THAN OR EQUAL TO +2.
SO BECAUSE OF THE EQUAL PART OF THIS, TWO IS INCLUDED
IN THE INTERVAL, AND THEREFORE WE MAKE A CLOSED POINT ON +2.
"X" IS ALSO LESS THAN TWO, SO WE HAVE AN ARROW
TO THE LEFT WHERE VALUES GET SMALLER THAN +2.
WE WOULD APPROACH NEGATIVE INFINITY.
SO HERE'S THE GRAPH OF THE INTERVAL,
AND THEN USING INTERVAL NOTATION,
WE HAVE NEGATIVE INFINITY TO +2.
BUT BECAUSE IT INCLUDES +2, WE USE A SQUARE BRACKET
HERE AND, AS ALWAYS FOR NEGATIVE INFINITY, A ROUNDED PARENTHESIS.
NOW LET'S TAKE A LOOK AT TWO COMPOUND INEQUALITIES.
HERE WE HAVE -3 IS LESS THAN OR EQUAL TO "X"
AND "X" IS LESS THAN 4.
SO NOTICE HOW THE INTERVAL IS BETWEEN -3 AND +4
WHERE IT DOES INCLUDE -3, BUT IT DOES NOT INCLUDE +4.
SO WE START AT -3 WITH A CLOSED POINT
BECAUSE IT INCLUDES -3.
AND THEN IT MOVES TO THE RIGHT UNTIL WE REACH +4,
AND IT DOES NOT INCLUDE +4, SO WE MAKE IT OPEN POINT HERE.
IT MIGHT ALSO BE HELPFUL TO THINK OF THIS
AS TWO INEQUALITIES CONNECTED WITH AND,
MEANING IF WE START IN THE MIDDLE WITH "X,"
WE CAN READ THIS FROM RIGHT TO LEFT
AS WELL AS FROM LEFT TO RIGHT.
WE CAN ALSO WRITE THIS AS "X" IS GREATER THAN OR EQUAL TO -3.
NOTICE HOW THE INEQUALITY IS STILL OPEN TOWARD THE "X."
AND "X" IS LESS THAN 4.
THESE TWO MEAN THE SAME THING, JUST WRITTEN A DIFFERENT WAY.
AND THEN USING INTERVAL NOTATION, THE INTERVAL
IS FROM -3 TO +4.
IT INCLUDES -3, SO WE HAVE A SQUARE BRACKET.
IT DOES NOT INCLUDE +4, SO WE HAVE
A ROUNDED PARENTHESIS.
SO WE SAY THE INTERVAL IS CLOSED ON -3 AND OPEN ON 4.
NEXT, WE HAVE "X" IS LESS THAN OR EQUAL TO -3
OR "X" IS GREATER THAN 4.
SO NOTICE HOW THE FIRST INTERVAL DOES INCLUDE -3,
SO WE HAVE A CLOSED POINT.
AND "X" IS LESS THAN OR EQUAL TO -3, SO WE HAVE THE INTERVAL
TO THE LEFT APPROACHING NEGATIVE INFINITY.
OR WE HAVE "X" IS GREATER THAN + 4.
THIS INTERVAL DOES NOT INCLUDE FOUR,
SO WE HAVE AN OPEN POINT HERE.
AND BECAUSE IT'S GREATER THAN FOUR,
THE INTERVALS TO THE RIGHT APPROACHING POSITIVE INFINITY.
SO BECAUSE WE HAVE TWO INTERVALS,
WE'LL HAVE TWO INTERVALS USING INTERVAL NOTATION.
THE INTERVAL ON THE LEFT, WE WOULD HAVE NEGATIVE INFINITY
TO -3, ROUNDED PARENTHESIS ON NEGATIVE INFINITY,
SQUARE BRACKET ON -3 BECAUSE IT INCLUDES -3.
THEN WE CAN SAY OR-- OR WE CAN ALSO USE "U"
FOR UNION--THEN WE HAVE THE INTERVAL FROM FOUR
TO INFINITY, ROUNDED PARENTHESIS ON FOUR
AS WELL AS INFINITY.
IT'S ROUNDED ON FOUR BECAUSE FOUR IS NOT INCLUDED
IN THIS INTERVAL.
OKAY. THAT'S GOING TO DO IT FOR THESE EXAMPLES.
I HOPE YOU FOUND THIS HELPFUL.