- WELCOME TO LINEAR INEQUALITIES IN TWO VARIABLES.
THE GOAL OF THIS VIDEO IS TO SOLVE LINEAR INEQUALITIES.
WHEN THE EQUAL SIGN IN A LINEAR EQUATION
IS REPLACED WITH AN INEQUALITY SIGN,
A LINEAR INEQUALITY IS FORMED.
ON THE LEFT WE SEE A LINEAR EQUATION,
AND ON THE RIGHT WE CAN SEE WE HAVE A LINEAR INEQUALITY.
THE SOLUTION FOR A LINEAR EQUATION CONSISTS OF ALL POINTS
(X,Y) THAT SATISFY THE EQUATION.
THE SOLUTION FORMS A LINE.
HOWEVER, WHEN WE HAVE A LINEAR INEQUALITY,
THE SOLUTION CONSISTS OF ALL THE POINTS
THAT SATISFY THE INEQUALITY,
BUT THE SOLUTION FORMS A REGION OR HALF PLANE
WITH THE LINE AS A BORDER.
AND THE LINE MAY OR MAY NOT BE INCLUDED IN THE SOLUTION
BASED UPON THE INEQUALITY SYMBOL.
HERE ARE THE STEPS TO GRAPHING THE SOLUTION
TO A LINEAR INEQUALITY.
STEP ONE, WE WILL GRAPH THE INEQUALITY AS IF IT WAS A LINE.
HOWEVER, IF THE INEQUALITY SYMBOL IS < OR >,
WE WILL MAKE A DASHED LINE,
BECAUSE THE LINE WILL NOT BE PART OF THE SOLUTION.
IF THE INEQUALITY SYMBOL IS A LESS THAN OR EQUAL
OR GREATER THAN OR EQUAL SIGN,
WE WILL MAKE A SOLID LINE,
BECAUSE THE LINE WILL BE PART OF THE SOLUTION.
NEXT, WE WILL TEST THE POINT ON EITHER SIDE OF THE LINE.
IF THE COORDINATES OF THE POINT SATISFY THE ORIGINAL INEQUALITY,
WE WILL SHADE THE SAME SIDE OF THE LINE AS THE POINT.
IF THE INEQUALITY IS FALSE,
WE WILL SHADE THE OTHER SIDE OF THE LINE.
THERE IS A SHORTCUT IF THE INEQUALITY IS SOLVED FOR Y.
YOU CAN SHADE ABOVE THE LINE
FOR > OR GREATER THAN OR EQUAL TO.
FOR < OR LESS THAN OR EQUAL TO, YOU WOULD SHADE BELOW THE LINE.
LET'S GO AHEAD AND GIVE IT A TRY.
LET'S GRAPH THE SOLUTION TO Y > 2X - 3.
SO STEP ONE, WE ARE GOING TO GRAPH THE LINE Y = 2X - 3,
BUT IT WILL BE A DASHED LINE,
BECAUSE Y IS NOT EQUAL TO 2X - 3.
IT'S ONLY GREATER THAN.
AND WE ALSO MAY WANT TO WRITE OUR SLOPE AS 2/1.
SO OUR Y INTERCEPT IS -3,
AND FROM HERE WE'LL GO UP TWO RIGHT ONE SEVERAL TIMES
TO FIND ADDITIONAL POINTS.
AND BEFORE WE SKETCH THIS,
WE HAVE TO REMEMBER IT'S GOING TO BE DASHED,
BECAUSE THE LINE IS NOT PART OF THE SOLUTION.
NOW, WE EITHER HAVE TO SHADE ABOVE OR BELOW THE LINE,
BECAUSE IT SAYS, ">," AND IT'S SOLVED FOR Y.
WE ARE GOING TO SHADE ABOVE OR TO THE LEFT,
BUT LET'S JUST PICK A TEST POINT TO MAKE SURE.
WE'LL PICK THE TEST POINT (0,0).
THAT'S USUALLY THE EASIEST POINT TO USE,
AS LONG AS IT'S NOT ON THE LINE.
SO LET'S REPLACE Y WITH ZERO AND X WITH ZERO.
SO WE'D HAVE 0 > 0 - 3 OR -3.
THIS IS TRUE, WHICH DOES VERIFY
WE SHOULD SHADE THE SAME SIDE OF THE LINE AS THE TEST POINT
OR THE SIDE.
SO THE REGION WE'LL SHADE IS THIS REGION HERE,
WHICH MEANS IF WE PICK ANY POINT IN THIS REGION,
IT WOULD SATISFY THE ORIGINAL INEQUALITY.
LET'S GRAPH THE SOLUTION TO 3X - 2Y = 12.
SO STEP ONE, WE WILL GRAPH THE LINE 3X - 2Y = 12.
BECAUSE IT'S GREATER THAN OR EQUAL TO, IT'LL BE A SOLID LINE.
NOW, THIS DOES NOT SOLVE FOR Y.
SO ON THIS PROBLEM I'M GONNA GO AHEAD AND JUST FIND
THE X AND THE Y INTERCEPTS
BY SETTING X = 0 AND Y = 0 IN COMPLETING THIS TABLE.
SO IF WE LET X = 0, WE WOULD HAVE -2Y = 12
DIVIDING BY -2, WE'D HAVE A Y INTERCEPT OF -6.
NOW, WE'LL SET Y = 0. SO THIS WOULD BE 0.
WE'D HAVE 3X = 12, DIVIDING BY 3, X = 4.
LET'S PLOT THESE TWO POINTS.
NOW, (0,-6) IS OFF THIS GRID BY ONE UNIT.
SO I'M GOING TO APPROXIMATE THE POINT TO BE THERE.
(4,0) WOULD BE HERE.
SO OUR LINE WILL PASS THROUGH THESE TWO POINTS.
I DID FIND ONE MORE POINT JUST TO MAKE MY GRAPH MORE ACCURATE.
MY LINE IS A LITTLE BIT OFF, BUT YOU GET THE IDEA.
BECAUSE THIS IS NOT SOLVED FOR Y,
LET'S GO AHEAD AND PICK A TEST POINT.
AGAIN, WE'LL CHOSE (0,0).
SO WE'LL SUB IN A ZERO FOR X AND A ZERO FOR Y.
SO THE LEFT SIDE WOULD BE 0,
AND THE QUESTION IS, IS TRUE OR FALSE:
IS ZERO GREATER THAN OR EQUAL TO 12,
AND THAT'S FALSE.
SO IF THIS POINT IS FALSE,
THIS ENTIRE HALF PLANE WOULD BE FALSE.
THEREFORE, WE WANT TO SHADE THE OTHER HALF OF THE LINE.
THIS IS A GREAT EXAMPLE OF A PROBLEM
WHERE WE DON'T SHADE ABOVE THE LINE,
BECAUSE WE SEE A GREATER THAN OR EQUAL TO INEQUALITY SYMBOL.
THAT ONLY WORKS WHEN IT'S SOLVED FOR Y.
SO I ALWAYS LIKE TO PICK A TEST POINT.
LET'S GO AHEAD AND TRY ONE MORE.
HERE WE HAVE X < -2.
SO WE ARE GOING TO GRAPH X = -2, BUT IT WILL BE DASHED,
BECAUSE THE LINE IS NOT PART OF THE SOLUTION.
NOW, THIS IS ONE OF THOSE WHAT I CALL SPECIAL LINES.
IT'S GOING TO BE EITHER HORIZONTAL OR VERTICAL,
AND IF YOU CAN'T REMEMBER,
I ALWAYS RECOMMEND MAKING A T TABLE.
THIS EQUATION STATES, "X HAS TO BE EQUAL TO -2."
IT DOES NOT INDICATE ANYTHING ABOUT Y.
SO Y CAN BE ANY VALUE YOU WISH, SO MAYBE 0 AND 1.
SO IF WE PLOT THOSE TWO POINTS,
WE'LL KNOW WHICH LINE TO GRAPH, (-2,0), (-2,1).
NOW, WE CAN EASILY SEE IT IS GOING TO BE A VERTICAL LINE,
BUT IT WILL BE DASHED.
AGAIN, WE'LL PICK A TEST POINT
TO SEE WHETHER WE SHOULD SHADE LEFT OR RIGHT.
WE'LL PICK THE POINT (0,0).
SO WE ONLY NEED THE X VALUE OF ZERO FOR THIS INEQUALITY,
AND WE HAVE 0 < -2, WHICH IS ACTUALLY FALSE.
SO THAT MEANS THE RIGHT SIDE IS FALSE.
THEREFORE, WE SHADE LEFT.
ANY POINT IN THIS REGION WOULD SATISFY THIS INEQUALITY.
I HOPE YOU FOUND THIS VIDEO USEFUL.
THANK YOU AND HAVE A GOOD DAY.