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- WELCOME TO A LESSON ON SOLVING
TWO-STEP LINEAR INEQUALITIES IN ONE VARIABLE.
THE GOALS ARE TO SOLVE TWO-STEP LINEAR INEQUALITIES
AS WELL AS EXPRESS THE SOLUTIONS AS AN INEQUALITY, A GRAPH,
AND USING INTERVAL NOTATION.
SO HERE'S A REVIEW OF THE THREE WAYS
THAT WE'LL EXPRESS THE SOLUTIONS.
HERE ARE SOME EXAMPLES OF INEQUALITY NOTATION.
HERE'S THE SAME INTERVAL GRAPH ON THE NUMBER LINE.
AND HERE'S THE SAME INTERVAL EXPRESSED USING
INTERVAL NOTATION.
IF NEEDED, YOU MAY WANT TO PAUSE THE VIDEO HERE AND REVIEW
THESE THREE WAYS TO EXPRESS THE SOLUTION OF AN INEQUALITY.
NOW THE STEPS TO SOLVE A LINEAR INEQUALITY
IN ONE VARIABLE ARE THE SAME IF WE WERE SOLVING
A LINEAR EQUATION EXCEPT IF WE MULTIPLY OR DIVIDE
BY A NEGATIVE WHEN ISOLATING THE VARIABLE,
WE MUST REVERSE THE INEQUALITY SYMBOL.
LOOKING AT THE INEQUALITY, -2 IS LESS THAN 4.
NOTICE IF WE DIVIDED BOTH SIDES BY -2,
THE LEFT SIDE WOULD BE +1, THE RIGHT SIDE WOULD BE -2
AND THEN, FOR THIS INEQUALITY TO BE TRUE, WE WOULD HAVE
TO REVERSE THIS INEQUALITY.
SO INSTEAD OF LESS THAN, IT WOULD BE GREATER THAN.
1 IS GREATER THAN -2, WHICH IS TRUE.
SO WE NEED TO BE SURE TO KEEP THIS RULE IN MIND
WHEN SOLVING OUR INEQUALITIES.
HERE WE HAVE 3X + 4 IS GREATER THAN OR EQUAL TO 10.
SO TO ISOLATE X ON THE LEFT SIDE,
WE WOULD START BY UNDOING THIS ADDITION
BY SUBTRACTING 4 ON BOTH SIDES.
THIS WOULD BE 0, SO WE HAVE 3X IS GREATER THAN
OR EQUAL TO 10 - 4 IS EQUAL TO 6.
AND SINCE 3X MEANS 3 x X, TO ISOLATE X,
WE'LL HAVE TO DIVIDE BOTH SIDES BY 3 AND WE'RE DIVIDING
BY A POSITIVE, SO WE DO NOT REVERSE THE INEQUALITY SYMBOL.
THIS WOULD BE 1X OR JUST X, GREATER THAN OR EQUAL TO +2.
SO HERE'S OUR SOLUTION AS AN INEQUALITY.
AND NOW WE'LL GO AHEAD AND GRAPH THIS INTERVAL
ON THE NUMBER LINE.
SO IF THIS 0, LET'S GO AHEAD AND CALL THIS 2.
X IS GREATER THAN OR EQUAL TO 2
AND, SINCE 2 IS IN THIS INTERVAL,
WE MAKE A CLOSED POINT ON 2.
THEN FOR VALUES GREATER THAN 2, WE'D HAVE AN ARROW TO THE RIGHT,
APPROACHING POSITIVE INFINITY.
NOW THE REASON I LIKE TO PUT THIS IN HERE IS IT MAKES
THE TRANSITION TO INTERVAL NOTATION VERY STRAIGHTFORWARD,
MEANING USING INTERVAL NOTATION WILL HAVE INTERVAL
FROM 2 TO INFINITY.
IT INCLUDES 2, SO WE HAVE A SQUARE BRACKET
TO THE LEFT OF 2 AND THEN WE HAVE A ROUNDED PARENTHESIS
FOR POSITIVE INFINITY.
WE ALWAYS USE A ROUNDED PARENTHESIS FOR POSITIVE
OR NEGATIVE INFINITY.
LOOKING AT OUR SECOND EXAMPLE, WE HAVE -2X - 1
IS GREATER THAN 9.
SO WE'LL START BY UNDOING THE SUBTRACTION.
SO WE'LL ADD TO BOTH SIDES.
THIS WOULD BE 0, SO WE HAVE
-2X IS GREATER THAN 9 + 1 IS 10.
NOW TO SOLVE FOR X, WE HAVE TO DIVIDE
BOTH SIDES BY -2.
AND SINCE WE'RE DIVIDING BY A NEGATIVE,
WE DO HAVE TO REVERSE THE INEQUALITY SYMBOL.
SO THIS WILL BE 1X OR JUST X.
THEN INSTEAD OF GREATER THAN, IT'S GOING TO BE LESS THAN.
10 DIVIDED BY -2 IS EQUAL TO -5.
SO HERE'S OUR SOLUTION AS AN INEQUALITY.
NOTICE HOW WE DID REVERSE THE INEQUALITY SYMBOL
BECAUSE WE DIVIDED BOTH SIDES BY -2.
NOW WE'LL GO AHEAD AND GRAPH THIS INTERVAL.
SO IF THIS IS 0, LET'S CALL THIS -5.
THIS INTERVAL DOES NOT INCLUDE -5,
SO NOW WE HAVE AN OPEN POINT ON -5.
THEN FOR VALUES LESS THAN -5, WE HAVE AN ARROW TO THE LEFT,
THIS TIME APPROACHING NEGATIVE INFINITY.
SO USING INTERVAL NOTATION, WE'LL HAVE NEGATIVE INFINITY
AND -5.
IT DOESN'T INCLUDE -5, SO WE HAVE A ROUNDED PARENTHESIS
HERE AS WELL AS HERE.
LET'S TAKE A LOOK AT TWO MORE EXAMPLES.
NOTICE HERE THESE EXAMPLES HAVE THE VARIABLE
ON THE RIGHT SIDE, BUT WE'LL FOLLOW THE SAME STEPS.
BUT NOW WE'LL JUST ISOLATE X ON THE RIGHT.
SO TO UNDO THE SUBTRACTION, WE'LL ADD 2 TO BOTH SIDES.
THIS WOULD BE 0, SO WE HAVE 12
GREATER THAN OR EQUAL TO, THIS IS -3X.
AND NOW TO SOLVE FOR X, WE HAVE TO DIVIDE BOTH SIDES BY A -3.
AGAIN, WE'RE DIVIDING BY A NEGATIVE,
SO WE'RE GOING TO REVERSE THE INEQUALITY SYMBOL.
SO HERE WE'LL HAVE -4 AND, INSTEAD OF GREATER THAN
OR EQUAL TO, WE'LL HAVE LESS THAN OR EQUAL TO X.
NOW THIS IS THE SOLUTION EXPRESSED AS AN INEQUALITY,
BUT WE CAN GO AHEAD AND REVERSE THE ORDER HERE.
THIS IS THE EQUIVALENT TO X IS GREATER THAN OR EQUAL TO -4.
NOTICE HOW IN BOTH CASES THE INEQUALITY SYMBOL
IS POINTING TOWARD THE -4 AND IT'S OPEN TOWARD THE X.
LET'S GO AHEAD AND LEAVE OUR ANSWER IN THIS FORM HERE.
AND NOW WE'LL GRAPH THIS.
X IS GREATER THAN OR EQUAL TO -4.
LET'S CALL THIS 0.
LET'S CALL THIS -4.
- 4 IS IN THIS INTERVAL, SO WE HAVE A CLOSED POINT
ON -4 AND THEN, FOR NUMBERS GREATER THAN -4,
WE HAVE AN ARROW TO THE RIGHT, APPROACHING POSITIVE INFINITY.
SO USING INTERVAL NOTATION, WE HAVE INTERVAL
FROM -4 TO INFINITY.
IT INCLUDES -4 WHERE WE SAY IT'S CLOSED ON -4,
SO WE USE A SQUARE BRACKET HERE AND A ROUNDED PARENTHESIS HERE.
OKAY, LET'S TAKE A LOOK AT OUR LAST EXAMPLE.
WE WANT TO ISOLATE THE X ON THE RIGHT SIDE,
SO WE NEED TO SUBTRACT 12 ON BOTH SIDES.
THIS IS GOING TO BE -20 IS GREATER THAN.
THIS WOULD BE 0, SO WE'RE LEFT WITH 5X ON THE RIGHT
AND NOW TO SOLVE FOR X, WE'LL DIVIDE BY 5 ON BOTH SIDES.
HERE WE'RE DIVING BY A +5, SO WE'RE NOT
GOING TO REVERSE INEQUALITY.
SO WE HAVE -4 IS GREATER THAN X, BUT AGAIN, IT'S MORE COMMON
TO REVERSE THE ORDER HERE.
NOTICE HOW THE INEQUALITY SYMBOL IS POINTING TOWARD THE X
AND OPEN TOWARD THE -4.
SO WE CAN REWRITE THIS AS X IS LESS THAN -4.
LET'S GO AHEAD AND GRAPH THIS.
- 4 IS NOT LESS THAN -4, SO -4 IS NOT IN THE INTERVAL.
SO WE HAVE AN OPEN POINT ON -4 AND,
FOR VALUES LESS THAN -4, WE HAVE AN ARROW TO THE LEFT,
APPROACHING NEGATIVE INFINITY.
SO USING INTERVAL NOTATION, WE HAVE NEGATIVE INFINITY,
WE HAVE A -4.
IT DOES NOT INCLUDE -4, SO WE USE A ROUNDED PARENTHESIS.
AND THAT'S GOING TO DO IT FOR THIS LESSON.