- WE WANT TO SOLVE AND GRAPH EACH INEQUALITY,
AND THEN EXPRESS THE SOLUTION USING INTERVAL NOTATION.
AND WE SOLVE THE LINEAR EQUALITY IN THE EXACT SAME
THAT WE SOLVE AN EQUATION,
EXCEPT WHEN SOLVING AN INEQUALITY,
IF WE MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER,
WE MUST REVERSE THE INEQUALITY SYMBOL.
LOOKING AT THE FIRST INEQUALITY, WE HAVE 2X + 8 > 16,
SO WE WANT TO ISOLATE X ON THE LEFT SIDE OF THE INEQUALITY.
SO WE NEED TO UNDO THIS ADDITION
AND THEN UNDO THIS MULTIPLICATION HERE.
SO WE'LL START BY UNDOING THE ADDITION,
SO WE'LL SUBTRACT 8 ON BOTH SIDES OF THE INEQUALITY.
SIMPLIFYING, WE HAVE 2X, THIS WOULD BE 0 >, 16 - 8
WOULD BE POSITIVE 8, AND AGAIN 2X MEANS 2 x X.
SO TO UNDO THIS MULTIPLICATION, WE'LL DIVIDE BOTH SIDES BY 2.
THEN WE'RE DIVIDING BY A POSITIVE,
THEREFORE WE DO NOT REVERSE THE INEQUALITY SYMBOL.
SO WHEN WE SIMPLIFY THIS, WE'LL HAVE X > POSITIVE 4.
NOW LET'S GO AHEAD AND GRAPH THIS.
NOTICE THAT 4 DOES NOT SATISFY THIS INEQUALITY,
BECAUSE 4 IS NOT > 4.
SO TO SHOW POSITIVE 4 IS NOT IN THIS INTERVAL,
WE MAKE AN OPEN POINT ON POSITIVE 4,
AND THEN NUMBERS > 4 WOULD BE TO THE RIGHT OF POSITIVE 4,
SO WE DRAW AN ARROW TO THE RIGHT.
AND AS WE CONTINUE TO MOVE RIGHT ON THE NUMBER LINE,
WE WOULD BE APPROACHING POSITIVE INFINITY,
AND KNOWING THIS IS GOING TO HELP WRITE THE INTERVAL
IN INTERVAL NOTATION.
THE INTERVAL IS FROM 4 TO INFINITY, NOT INCLUDING 4.
SO AN INTERVAL NOTATION WILL HAVE 4 AND INFINITY.
AND TO SHOW IT DOES NOT INCLUDE 4,
WE MAKE A ROUNDED PARENTHESIS HERE.
AND THEN FOR INFINITY, WE ALWAYS USE A ROUNDED PARENTHESIS.
SO NOW WE HAVE EXPRESSED THE SOLUTION AS AN INEQUALITY,
AS A GRAPH, AND AN INTERVAL NOTATION.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
AGAIN OUR GOAL IS TO ISOLATE X
ON THE LEFT SIDE OF THE INEQUALITY.
REMEMBER, THESE ARE CONNECTED BY MULTIPLICATION HERE,
SO WE FIRST WANT TO UNDO THIS POSITIVE 4.
WELL, TO UNDO POSITIVE 4,
WE CAN SUBTRACT 4 ON BOTH SIDES OF THE INEQUALITY.
4 - 4 = 0, THAT'S WHY WE DID THAT,
SO NOW WE HAVE -4X IS GREATER THAN OR EQUAL TO, -2 - 4 = -6.
AND AGAIN BECAUSE -4X MEANS -4 TIMES X,
WE'LL HAVE TO DIVIDE BOTH SIDES BY -4 TO SOLVE FOR X.
NOW AGAIN NOTICE WE'RE DIVIDING BY A NEGATIVE,
SO WE DO HAVE TO REVERSE THE INEQUALITY SYMBOL.
SO ON THE LEFT, WE HAVE X,
AND THEN INSTEAD OF GREATER THAN OR EQUAL TO,
WE'LL HAVE LESS THAN OR EQUAL TO,
AND THEN -6 DIVIDED BY -4 WOULD SIMPLIFY TO POSITIVE 3/2.
AND IF IT'S HELPFUL, WE CAN CONVERT THIS TO A DECIMAL.
REMEMBER, 3/2 MEANS 3 DIVIDED BY 2 = 1.5.
AND AGAIN LOOKING AT THIS INTERVAL,
NOTICE THAT 3/2 WOULD SATISFY THIS INEQUALITY
BECAUSE 3/2 = 3/2.
SO WHEN WE PLOT 3/2 THIS TIME, MAKE A CLOSED POINT.
AND 3/2 AGAIN IS BETWEEN 1 AND 2, SO LET'S SAY IT'S RIGHT HERE.
SO WE'LL MAKE A CLOSED POINT ON 3/2,
AND THEN FOR THE NUMBERS LESS THAN 3/2,
WE'LL MAKE AN ARROW TO THE LEFT.
AND AGAIN AS WE MOVE FURTHER AND FURTHER LEFT ON THE NUMBER LINE,
WE WOULD BE APPROACHING - INFINITY.
SO USING INTERVAL NOTATION,
WE HAVE THE INTERVAL FROM 3/2 TO -INFINITY INCLUDING 3/2,
SO WE USE -INFINITY 3/2,
A ROUNDED PARENTHESIS ON - INFINITY.
BUT TO SHOW THAT 3/2 IS INCLUDED IN THE INTERVAL,
WE MAKE A SQUARE BRACKET AROUND 3/2.
SO AGAIN WE HAVE THE SOLUTION EXPRESSED THREE WAYS,
AS AN INEQUALITY, AS A GRAPH, AND USING INTERVAL NOTATION.
WE'LL TAKE A LOOK AT MORE EXAMPLES IN THE NEXT VIDEO.