- WE WANT TO SOLVE AND GRAPH EACH INEQUALITY,
AND THEN EXPRESS THE SOLUTION USING INTERVAL NOTATION.
THE METHOD FOR SOLVING A LINEAR INEQUALITY IS THE SAME
AS SOLVING A LINEAR EQUATION,
EXCEPT WHEN SOLVING A LINEAR INEQUALITY,
IF WE MULTIPLY OR DIVIDE BY A NEGATIVE,
WE MUST REVERSE THE INEQUALITY SYMBOL.
LOOKING AT OUR FIRST EXAMPLE,
NOTICE WE HAVE VARIABLE TERMS ON BOTH SIDES OF THE INEQUALITY.
SO THE FIRST STEP IS TO GET THE VARIABLE TERMS ON ONE SIDE.
AND SINCE WE NORMALLY PREFER TO HAVE THE VARIABLES
ON THE LEFT SIDE OF AN INEQUALITY, TO UNDO POSITIVE 3X,
WE'LL SUBTRACT 3X ON BOTH SIDES OF THE INEQUALITY.
SO NOW WE HAVE 2X - 3 = -1X OR - X - 6 >,
WELL, 3X - 3X = 0, THAT'S WHY WE DID THAT, SO WE HAVE -11.
AND AGAIN, IT MIGHT BE HELPFUL TO THINK OF THIS AS -1X,
SO WE NEED TO UNDO THE SUBTRACTION HERE
AS WELL AS THE MULTIPLICATION BETWEEN THE -1 AND X.
AND WE ALWAYS UNDO THE ADDITION OR SUBTRACTION FIRST.
SO TO UNDO -6, WE'LL ADD 6 TO BOTH SIDES, SO WE HAVE -1X.
THIS WOULD BE 0 IS GREATER THAN, -11 + 6 = -5.
AND NOW TO UNDO THIS MULTIPLICATION,
WE'LL DIVIDE BOTH SIDES BY -1.
AND SINCE WE'RE DIVIDING BY A NEGATIVE,
WE DO HAVE TO REVERSE THE INEQUALITY SYMBOL,
SO HERE WE'LL HAVE POSITIVE 1X OR X.
AND IF THIS IS GREATER THAN AND THEN WE REVERSE IT,
WE'LL HAVE X <, AND THEN -5 DIVIDED BY -1 = POSITIVE 5.
SO OUR SOLUTION IS X IS LESS THAN POSITIVE 5,
SO TO GRAPH THIS,
SINCE 5 IS NOT IN THIS INTERVAL, WE'LL MAKE AN OPEN POINT ON 5.
AND THEN FOR VALUES LESS THAN 5, WE HAVE AN INTERVAL TO THE LEFT.
AS WE MOVE FURTHER AND FURTHER TO THE LEFT,
WE WOULD BE APPROACHING -INFINITY,
KNOWING THIS IS HELPFUL TO WRITE THIS INTERVAL
USING INTERVAL NOTATION.
WE WANT THE INTERVAL FROM 5 TO -INFINITY, NOT INCLUDING 5,
SO WE'LL HAVE TO INCLUDE A -INFINITY HERE AND A 5.
AND WE USE A ROUNDED PARENTHESIS TO THE LEFT OF -INFINITY,
AND THEN TO SHOW THAT IT DOES NOT INCLUDE POSITIVE 5,
WE MAKE ANOTHER ROUNDED PARENTHESIS HERE.
SO NOW WE'VE EXPRESSED THE SOLUTION THREE WAYS,
AS AN INEQUALITY, AS A GRAPH, AND USING INTERVAL NOTATION.
LOOKING AT OUR SECOND EXAMPLE,
AGAIN, WE HAVE VARIABLE TERMS ON BOTH SIDES OF INEQUALITY,
SO WE NEED TO EITHER SUBTRACT 5X ON BOTH SIDES OR SUBTRACT 8X
ON BOTH SIDES.
IF WE PREFER TO HAVE THE VARIABLE ON THE LEFT SIDE,
WE WOULD HAVE TO SUBTRACT 5X ON BOTH SIDES.
SO 8X - 5X = 3X - 4 GREATER THAN OR EQUAL TO, THIS WOULD BE 0,
AND WE HAVE 14.
NOW WE NEED TO UNDO THE SUBTRACTION
AND THIS MULTIPLICATION,
AND WE ALWAYS UNDO THE SUBTRACTION FIRST,
SO ADD 4 TO BOTH SIDES.
SO WE'LL HAVE 3X SHOULD BE 0 GREATER THAN OR EQUAL TO 18,
AND THE LAST STEP IS TO DIVIDE BOTH SIDES BY 3.
NOW WE'RE DIVIDING BY A POSITIVE 3,
SO WE DO NOT REVERSE THE INEQUALITY SYMBOL.
SO WE HAVE X GREATER THAN OR EQUAL TO 18 DIVIDED BY 3
= POSITIVE 6.
AND NOTICE IN THIS CASE, 6 DOES SATISFY THIS INEQUALITY,
BECAUSE 6 IS GREATER THAN OR EQUAL TO 6.
SO TO SHOW THIS GRAPHICALLY, WE MAKE IT A CLOSED POINT ON 6.
AND THEN FOR NUMBERS GREATER THAN 6,
WE DRAW AN ARROW TO THE RIGHT
WHICH WOULD BE APPROACHING POSITIVE INFINITY.
THEREFORE FOR INTERVAL NOTATION,
WE HAVE THE INTERVAL FROM 6 TO INFINITY.
BUT TO SHOW THAT IT INCLUDES 6, WE MAKE A SQUARE BRACKET HERE,
AND THEN FOR INFINITY, WE ALWAYS USE A ROUNDED PARENTHESIS.
OKAY, I HOPE YOU FOUND THIS HELPFUL.