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- WELCOME TO A LESSON IN WHICH YOU'LL BE GIVEN
SEVERAL INTERVALS EXPRESSED USING INTERVAL NOTATION
AND THEN ASKED TO GRAPH THE INTERVAL AND EXPRESS
THE INTERVAL USING AN INEQUALITY.
SO LOOKING AT OUR FIRST INTERVAL HERE, WE HAVE INTERVAL
FROM -2 TO POSITIVE INFINITY, BUT BECAUSE OF THIS SQUARE
BRACKET HERE, THIS INDICATES THAT -2 IS IN THE INTERVAL
WHERE THE INTERVAL IS CLOSED ON -2.
SO TO GRAPH THIS, WE MAKE A CLOSED POINT
ON -2 TO INDICATE -2 IS IN THE INTERVAL,
AND THEN APPROACHING POSITIVE INFINITY MEANS WE'LL GRAPH
TO THE RIGHT, AGAIN, APPROACHING POSITIVE INFINITY.
AND SO NOW TO EXPRESS THIS USING INEQUALITY, WE WOULD SAY
"X" IS GREATER THAN OR EQUAL TO -2.
AGAIN, GREATER THAN BECAUSE WE'RE GRAPHING
TO THE RIGHT AND EQUAL TO BECAUSE -2 IS INCLUDED
IN THE INTERVAL OR BECAUSE OF THE CLOSED POINT
OR USING INTERVAL NOTATION BECAUSE OF THE SQUARE BRACKET.
LOOKING AT THE SECOND EXAMPLE, WE HAVE THE INTERVAL
FROM -2 TO NEGATIVE INFINITY.
NOTICE HOW IT DOES NOT INCLUDE -2, AND WE KNOW THIS
BECAUSE OF THE ROUNDED PARENTHESIS.
SO TO GRAPH THIS, WE MAKE AN OPEN POINT ON -2,
AND THEN BECAUSE WE'RE APPROACHING NEGATIVE INFINITY,
WE GRAPH TO THE LEFT.
SO USING INEQUALITY, WE CAN SAY THAT "X"
IS LESS THAN -2.
NOTICE HOW WE DID NOT SAY LESS THAN OR EQUAL TO
BECAUSE -2 IS NOT INCLUDED IN THE INTERVAL.
NOW, TAKE A LOOK AT TWO COMPOUND INEQUALITIES.
HERE WE HAVE THE INTERVAL FROM - 3 TO +4 WHERE -3
IS INCLUDED BECAUSE OF THE SQUARE BRACKET
AND +4 IS NOT INCLUDED BECAUSE OF THE ROUNDED PARENTHESIS.
SO TO GRAPH THIS, WE'D MAKE A CLOSED POINT
ON -3 BECAUSE IT'S INCLUDED IN THE INTERVAL.
WE'D MAKE AN OPEN POINT ON +4, AND THEN GRAPH
ALL THE REAL NUMBERS BETWEEN THESE FROM -3 TO +4.
NOW LET'S WRITE THIS AS AN INEQUALITY.
THIS IS ACTUALLY THE INTERSECTION
OF TWO INEQUALITIES, SO WE'LL HAVE TWO INEQUALITIES
CONNECTED WITH "AND".
WE CAN SAY "X" IS GREATER THAN OR EQUAL TO -3
AND "X" ALSO HAS TO BE LESS THAN 4.
THESE BOTH HAVE TO BE TRUE TO DEFINE THIS INTERVAL.
BECAUSE THIS IS AN "AND" OR THE INTERSECTION
OF TWO INEQUALITIES, WE CAN ALSO WRITE THIS
A DIFFERENT WAY WHERE WE PUT "X" IN THE MIDDLE AND WRITE
THIS INEQUALITY IN THE OPPOSITE DIRECTION.
WE CAN SAY "X" IS GREATER THAN OR EQUAL TO -3
AND "X" IS LESS THAN 4.
SO BECAUSE THIS IS AN AND COMPOUND INEQUALITY,
WE CAN ALSO WRITE IT LIKE THIS, AND WE CAN ALSO SAY
-3 LESS THAN OR EQUAL TO "X" AND "X" IS LESS THAN 4.
THESE ARE BOTH EQUIVALENT, JUST WRITTEN A DIFFERENT WAY.
AND NOW FOR THE LAST EXAMPLE, HERE WE HAVE THE INTERVAL FROM
-3 TO NEGATIVE INFINITY WHERE, BECAUSE OF THE BRACKET,
IT INCLUDES -3.
LET'S GO AHEAD AND GRAPH THIS PART FIRST.
CLOSED POINT ON -3 APPROACHING NEGATIVE INFINITY,
SO WE GRAPH TO THE LEFT.
THIS "U" HERE MEANS UNION.
WE CAN ALSO SAY OR THE INTERVAL FROM FOUR TO INFINITY WHERE,
BECAUSE OF THIS ROUNDED PARENTHESIS,
FOUR IS NOT INCLUDED IN THE INTERVAL.
SO WE MAKE AN OPEN POINT ON FOUR, AND WE GRAPH
TO THE RIGHT APPROACHING POSITIVE INFINITY.
AND NOW FOR THE INEQUALITY, WE'LL HAVE "X" IS LESS THAN
OR EQUAL TO -3 OR "X" IS GREATER THAN +4.
WE USE THE WORD OR TO INDICATE THAT IF "X" IS IN EITHER
OF THESE INTERVALS, THEY WOULD FALL INTO THE GRAPHED INTERVAL.
OKAY. THAT'S GOING TO DO IT FOR THESE EXAMPLES.
I HOPE YOU FOUND THIS HELPFUL.