- WELCOME TO A PRESENTATION ON SOLVING LINEAR INEQUALITIES
IN ONE VARIABLE.
LET'S FIRST TALK ABOUT WHAT AN INEQUALITY LOOKS LIKE.
WE CAN SEE HERE THAT LINEAR EQUATIONS
AND LINEAR INEQUALITIES LOOK VERY SIMILAR.
HOWEVER, LINEAR EQUATIONS HAVE EQUAL SIGNS,
AND LINEAR INEQUALITIES HAVE INEQUALITY SIGNS.
ANY VALUE THAT MAKES AN INEQUALITY TRUE
IS CALLED A SOLUTION.
WHEN ALL SOLUTIONS OF AN INEQUALITY ARE FOUND,
WE SAY THAT WE HAVE SOLVED THE INEQUALITY.
WHEN WE GO TO SOLVE INEQUALITIES, WE'LL FIND
THAT WE HAVE AN INFINITE NUMBER OF SOLUTIONS IN MOST CASES,
WHEREAS WHEN YOU SOLVE A LINEAR EQUATION,
YOU EXPECT JUST ONE SOLUTION.
SO BECAUSE THERE'S AN INFINITE NUMBER OF SOLUTIONS
WHEN YOU'RE SOLVING A LINEAR INEQUALITY,
THE SOLUTIONS CAN BE EXPRESSED IN SEVERAL WAYS.
YOU CAN USE INEQUALITIES.
YOU CAN USE A GRAPH.
OR YOU CAN USE INTERVAL NOTATION.
SO, FOR EXAMPLE, IF WE HAVE THE INEQUALITY X < 2,
ANY VALUE THAT'S LESS THAN 2 WOULD BE A SOLUTION
TO THIS INEQUALITY.
THE GRAPH OF THAT, BECAUSE OF THIS INEQUALITY
SYMBOL, WOULD HAVE AN OPEN CIRCLE ON TWO
AND AN ARROW TO THE LEFT SHOWING ANY NUMBER
THAT WE'VE GRAPHED HERE WOULD SATISFY THIS INEQUALITY.
THIS CIRCLE IS OPEN BECAUSE TWO IS NOT A SOLUTION
TO THIS INEQUALITY.
WE SHOULD ALSO NOTE THAT AS WE MOVE TO THE LEFT
ON THE NUMBER LINE, WE ARE APPROACHING
NEGATIVE INFINITY.
AND I MENTION THAT BECAUSE IT'S EASY NOW TO GO FROM
THE GRAPH TO WHAT'S CALLED INTERVAL NOTATION.
ALL THESE THREE COMMUNICATE THE SAME INFORMATION.
YOU TAKE A LOOK AT THE SECOND INEQUALITY,
X IS LESS THAN OR EQUAL TO THREE,
THE LESS THAN OR EQUAL SYMBOL TRANSLATES INTO A CLOSED POINT
BECAUSE NOW THREE IS INCLUDED IN THAT SOLUTION.
IF YOU LOOK AT THE INTERVAL NOTATION,
TO SHOW THAT THREE IS INCLUDED, WE USE A SQUARE BRACKET.
AND THESE NEXT TWO EXAMPLES ARE SIMILAR,
EXCEPT NOW WE HAVE GREATER THAN AND GREATER THAN OR EQUAL TO,
SO NOW OUR ARROW GOES TO THE RIGHT,
WHICH MEANS WE'D BE APPROACHING POSITIVE INFINITY WHICH SHOWS UP
IN OUR INTERVAL NOTATION.
NOW, THE GOOD NEWS IS, TO SOLVE LINEAR INEQUALITIES,
WE SOLVE THEM JUST LIKE EQUATIONS,
EXCEPT IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE WHEN SOLVING
FOR THE VARIABLE,
YOU MUST REVERSE THE INEQUALITY SYMBOL.
HERE'S KIND OF AN EXPLANATION OF THAT.
YOU MAY WANT TO PAUSE AND TAKE A LOOK AT THIS.
ESSENTIALLY WHAT I'VE SHOWN HERE IS IF YOU ADD A POSITIVE NUMBER
OR SUBTRACT A POSITIVE NUMBER FROM BOTH SIDES IN INEQUALITY,
IT'S STILL A TRUE STATEMENT.
IF YOU MULTIPLY IT BY A POSITIVE OR DIVIDE BY A POSITIVE,
THE RESULTING INEQUALITY IS STILL TRUE.
HOWEVER, IF NOW WE CHANGE IT AND MULTIPLY IT BY A NEGATIVE
ON BOTH SIDES, WE GET A FALSE STATEMENT.
SO IN ORDER TO MAKE IT TRUE, WE HAVE TO REVERSE
THE INEQUALITY.
AND THE SAME THING IS TRUE IF WE DIVIDE BOTH SIDES BY A NEGATIVE.
WE HAVE TO REMEMBER TO REVERSE THE DIRECTION OF THE INEQUALITY
TO MAINTAIN THE TRUE STATEMENT.
OKAY. LET'S GO AHEAD AND TRY SOME OF THESE.
WE'RE GOING TO SOLVE THESE JUST LIKE THEY WERE EQUATIONS,
EXCEPT REMEMBERING IF WE MULTIPLY OR DIVIDE IT
BY A NEGATIVE, WE HAVE TO REVERSE THE SYMBOL.
SO THE FIRST STEP HERE WOULD BE TO ADD THREE TO BOTH SIDES.
SO WE'D HAVE -15 DIVIDED BY +5.
SO HERE IS OUR SOLUTION STATED AS AN INEQUALITY.
LET'S GO AHEAD AND GRAPH THIS.
"X" IS GREATER THAN OR EQUAL TO -3,
SO BECAUSE OF THIS INEQUALITY SYMBOL,
IT WOULD BE A CLOSED POINT.
AND BECAUSE IT'S GREATER THAN OR EQUAL TO,
WE WOULD GRAPH TO THE RIGHT.
AGAIN, AS WE MOVE TO THE RIGHT, WE'RE APPROACHING
POSITIVE INFINITY.
THIS WOULD BE OUR GRAPH.
IF WE WANT TO USE INTERVAL NOTATION,
OUR INTERVAL IS FROM -3 TO POSITIVE INFINITY.
IT INCLUDES -3, SO WE'D USE A SQUARE BRACKET.
AND FOR INFINITY, WE ALWAYS USE A ROUNDED PARENTHESIS.
SO THAT'S PRETTY MUCH IT.
WE'VE EXPRESSED THE ANSWER THREE WAYS, USING INEQUALITIES,
USING A GRAPH, AND INTERVAL NOTATION.
LET'S GO AHEAD AND TRY ANOTHER.
SO, AGAIN, WE'LL SOLVE THIS LIKE IT WAS AN EQUATION,
EXCEPT REMEMBERING THAT ONE ADDITIONAL RULE.
SO WE'LL CLEAR OUR PARENTHESES.
LET'S GET "X" ON ONE SIDE, SO WE'LL SUBTRACT "X"
ON BOTH SIDES.
AT THE SAME TIME, I'LL GO AHEAD AND GET RID OF THIS EIGHT
BY SUBTRACTING EIGHT ON BOTH SIDES.
SO THE RESULT OF THESE TWO STEPS WOULD BE -3X--THAT'S GONE--
IS GREATER THAN-- THAT'S GONE-- -28.
NOW, WE'LL DIVIDE BY -3-- HOWEVER, REMEMBER,
WHEN YOU DIVIDE BY A NEGATIVE, WE DO HAVE TO REVERSE
THE INEQUALITY SYMBOL.
SO WE'RE GOING TO HAVE "X" IS LESS THAN 28/3.
FOR THE PURPOSE OF GRAPHING THIS,
WE MAY WANT TO REWRITE THIS AS "X" IS LESS THAN 9 1/3.
OKAY. LET'S GRAPH THIS.
LET'S SEE, OUR 9 1/3 IS THERE BECAUSE
IT READS "X" IS LESS THAN 9 1/3, NOT EQUAL TO,
WE'D HAVE AN OPEN CIRCLE ON 9 1/3.
AND IN THIS CASE--OR IT WOULD GO TO THE LEFT BECAUSE IT'S LESS
THAN APPROACHING NEGATIVE INFINITY.
SO USING INTERVAL NOTATION, WE CAN SEE
WE HAVE A NEGATIVE INFINITY TO 9 1/3, AND THEY'RE BOTH--
AND 9 1/3, AND THEY'RE BOTH ROUNDED PARENTHESES.
LET'S NOW TAKE A LOOK AT HOW WE CAN SOLVE LINEAR INEQUALITIES
USING OUR GRAPHING CALCULATORS.
HERE ARE THE STEPS.
WE'LL APPLY IT TO THIS EXAMPLE DOWN HERE.
STEP ONE IS WE WANT TO GRAPH THE LEFT SIDE AS Y1,
SO THIS WOULD BE OUR Y1.
GRAPH THE RIGHT SIDE AS Y2, SO THIS WOULD BE Y2.
SO IF WE WANTED TO, WE COULD REWRITE THIS
AS WE WANT TO KNOW WHEN Y1 IS LESS THAN OR EQUAL TO Y2.
SO GRAPHICALLY, WHAT WE'RE LOOKING FOR
IS FOR WHICH "X" VALUES IS Y1 GOING TO BE EITHER BE LOW
OR EQUAL TO Y2.
SO WHAT WE'RE GOING TO DO THEN IS GRAPH THEM,
FIND THE POINT OF INTERSECTION, AND THEN, BASED UPON
THIS INEQUALITY, WE'LL DETERMINE THE SOLUTION.
SO LET'S GET OUR GRAPHING CALCULATORS OUT.
AS YOU CAN SEE, I'VE ALREADY TYPED Y1 AND Y2
INTO THE CALCULATOR.
LET'S GO AHEAD AND PRESS GRAPH.
NOW, SINCE THIS LINE HERE IS GOING UP FROM LEFT TO RIGHT,
I KNOW THIS IS Y1 BECAUSE IT HAS A POSITIVE SLOPE,
AND THIS IS Y2.
AGAIN, WE WANT TO KNOW WHERE Y1 IS BELOW Y2.
THIS PIECE HERE REPRESENTS OUR SOLUTION.
SO WE WANT TO KNOW FOR WHICH "X" VALUES
IS THIS LINE BELOW THIS ONE.
LET'S START BY FINDING THAT POINT OF INTERSECTION.
WE HIT 2ND, CALC, SELECT OPTION FIVE,
AND THEN ENTER THREE TIMES.
SO WE CAN SEE THEY'RE EQUAL WHEN "X" = 2.
AND THEN TO THE LEFT IS WHEN Y1 IS BELOW Y2.
SO OUR SOLUTION WILL BE, WHEN "X" IS LESS THAN
OR EQUAL TO TWO.
SO YOU MAY WANT TO CHECK YOUR SOLUTIONS GRAPHICALLY
BECAUSE, AS YOU CAN SEE, WE CAN CHECK OUR SOLUTION
WITHOUT DOING ANY ALGEBRAIC MANIPULATION.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.