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- WELCOME TO TWO EXAMPLES ON HOW TO SOLVE A LINEAR INEQUALITY
IN ONE VARIABLE BY GRAPHING ON THE COORDINATE PLANE.
A SOLUTION TO AN INEQUALITY CONTAINS ALL THE VALUES
THAT SATISFY THE INEQUALITY.
WE MOST OFTEN SOLVE INEQUALITIES ALGEBRAICALLY
BUT IN THIS VIDEO, WE WILL SOLVE LINEAR INEQUALITIES
IN ONE VARIABLE BY GRAPHING.
WE USE A TECHNIQUE CALLED THE INTERSECTION METHOD
WHICH MEANS IF WE HAVE A LINEAR INEQUALITY
IN THE FORM OF F OF X IS GREATER THAN G OF X
OR SOMETHING SIMILAR TO THIS
WITH A DIFFERENT INEQUALITY SYMBOL
WE'LL LET THE LEFT SIDE BE EQUAL TO Y1.
THE RIGHT SIDE BE EQUAL TO Y2,
AND THEN WE'LL GRAPH Y1 AND Y2 ON THE SAME COORDINATE PLANE.
ONCE WE DO THIS, WE'LL USE THE X VALUE
OF THE POINT OF INTERSECTION
TO DETERMINE WHICH X VALUES ACTUALLY SATISFY THE INEQUALITY.
SO GOING BACK TO OUR FIRST EXAMPLE,
WE'RE GOING TO LET THE LEFT SIDE BE EQUAL TO Y1,
SO Y1 = 2X - 3 AND Y2 = 5.
WE WANT TO KNOW WHEN Y1 IS LESS THAN Y2,
WHICH WOULD BE WHEN Y1 IS GRAPHICALLY BELOW Y2.
TO DETERMINE THESE X VALUES
WE'LL ACTUALLY USE THE INTERSECTION OF THE TWO LINES
AS REFERENCE.
LET'S GO AHEAD AND ENTER Y1 AND Y2.
WE'LL PRESS Y = Y1 IS 2X - 3, ENTER AND Y2 IS 5.
IT'S IMPORTANT THAT WE CAN TELL THESE TWO LINES APART
SO FOR Y2 WE'RE GOING TO GO TO THE LEFT OF Y2,
PRESS ENTER ONCE, AND NOW Y2 WILL BE A THICK LINE.
SO AGAIN, OUR GOAL HERE IS TO FIND FOR WHICH X VALUES Y1
OR THE THIN LINE IS BELOW Y2 WITH A THICK LINE.
TO MAKE SURE WE HAVE THE STANDARD WINDOW
LET'S PRESS ZOOM 6.
THERE'S Y1
THERE'S Y2.
LET'S CALCULATE THIS POINT OF INTERSECTION.
WE'LL PRESS 2nd TRACE OPTION 5, ENTER THREE TIMES.
THE POINT OF INTERSECTION IS THE POINT 4, 5.
BUT NOTICE HOW Y1 OR THE THIN LINE IS BELOW Y2
WITH A THICK LINE WHEN X IS LESS THAN 4 OR ON THE LEFT SIDE OF 4
MEANING THIS PIECE OF THE LINE HERE.
SO THAT'S OUR SOLUTION.
X IS LESS THAN 4.
WE DIDN'T INCLUDE 4 BECAUSE THAT'S WHEN Y1 WAS EQUAL TO Y2.
WE ONLY HAVE LESS THAN, NOT LESS THAN OR EQUAL TO.
BUT SOMETIMES WE'RE ALSO ASKED TO EXPRESS THIS SOLUTION
ON THE NUMBER LINE AS WELL AS USING INTERVAL NOTATION.
SO LET'S GO AHEAD AND SHOW THAT AS WELL.
IF WE WANT TO GRAPH, X IS LESS THAN 4.
WE WOULD HAVE A NUMBER LINE,
WE WOULD HAVE AN OPEN POINT ON 4,
AND BECAUSE X IS LESS THAN 4 WE'D HAVE AN ARROW TO THE LEFT.
AS WE MOVE TO THE LEFT,
WE WOULD ACTUALLY BE APPROACHING NEGATIVE INFINITY,
WHICH IS HELPFUL
BECAUSE IF WE WANTED TO EXPRESS OUR SOLUTION
USING INTERVAL NOTATION WE WOULD HAVE THE OPEN INTERVAL
FROM NEGATIVE INFINITY TO 4.
SO WE'VE EXPRESSED THE SOLUTION THREE WAYS, USING AN INEQUALITY,
USING A GRAPH ON THE NUMBER LINE,
AND USING INTERVAL NOTATION.
LET'S GO AND TAKE A LOOK AT A SECOND EXAMPLE.
HERE WE HAVE 2 IS GREATER THAN OR EQUAL TO 12 - 5X.
SO WE'LL LET Y1 BE EQUAL TO 2.
AND Y2 IS GOING TO BE EQUAL TO 12 - 5X.
SO WE WANT TO KNOW GRAPHICALLY
WHEN WOULD Y1 BE GREATER THAN OR EQUAL TO Y2,
WHICH WOULD BE WHEN Y1 IS ABOVE OR ON Y2.
SO IN THIS CASE, WE WILL INCLUDE THE POINT OF INTERSECTION.
LET'S GO BACK TO THE CALCULATOR, PRESS Y EQUALS,
CLEAR OUT THE OLD EQUATIONS AND TYPE IN THE NEW EQUATIONS.
Y1 IS 2, Y2 IS 12 - 5X.
AND AGAIN, IT'S IMPORTANT THAT WE TELL THESE 2 LINES APART,
SO WE'LL GO TO THE FAR LEFT OF Y2, PRESS ENTER.
SO Y2 WILL BE THE THICK LINE
AND BEFORE WE PRESS GRAPH
WE WANT TO KNOW WHEN Y1 IS GREATER THAN OR EQUAL TO Y2
OR WHEN THE THIN LINE IS EITHER ON OR ABOVE THE THICK LINE.
SO LET'S GO AHEAD AND PRESS GRAPH.
NOTICE HOW THE THIN LINE IS ABOVE THE THICK LINE
ON THE RIGHT SIDE OF THE SCREEN
SO LET'S FIND THIS POINT OF INTERSECTION.
WE'LL PRESS 2nd TRACE, OPTION 5, ENTER THREE TIMES.
THE POINT OF INTERSECTION IS THE POINT 2.2.
SO BECAUSE WE WANT TO KNOW
WHEN Y1 IS GREATER THAN OR EQUAL TO Y2
OR WHEN THE THIN LINE IS EQUAL TO OR ABOVE THE THICK LINE
OUR SOLUTION IS X IS GREATER THAN OR EQUAL TO 2.
LET'S GO AND SHOW THE GRAPH OF THIS ON THE NUMBER LINE.
THIS TIME WE INCLUDE 2
SO WE'LL MAKE A CLOSED POINT ON 2
AND X IS GREATER THAN OR EQUAL TO 2
SO THE ARROW GOES TO THE RIGHT APPROACHING POSITIVE INFINITY.
SO USING INTERVAL NOTATION WE'D HAVE THE INTERVAL
THAT'S CLOSED ON 2 TO POSITIVE INFINITY.
IT'S ALWAYS OPEN ON INFINITY.
SO AGAIN, WE'VE EXPRESSED THIS SOLUTION IN THREE WAYS,
USING AN INEQUALITY, USING THE NUMBER LINE,
AND USING INTERVAL NOTATION.
I HOPE YOU FOUND THESE TWO EXAMPLES HELPFUL.