- WE WANT TO GRAPH Y IS GREATER THAN OR EQUAL TO -2/3X + 4.
TO GRAPH THIS, WE WANT TO GRAPH THE REGION
WHERE ALL THE POINTS WOULD HAVE ORDERED PAIRS
THAT SATISFY THE GIVEN INEQUALITY.
TO DO THIS, WE TREAT THIS AS A LINE,
AND WE'RE GOING TO GRAPH Y = -2/3X + 4.
THIS IS ACTUALLY GOING TO BE THE BOUNDARY FOR THE REGION
THAT WE'LL END UP SHADING.
BUT THERE'S ONE MORE THING TO MENTION
BEFORE WE GRAPH THIS LINE.
BASED UPON THIS GIVEN INEQUALITY SYMBOL,
THE LINE IS EITHER GOING TO BE DASHED OR SOLID.
IF WE HAVE LESS THAN OR EQUAL TO OR GREATER THAN OR EQUAL TO,
WE'RE GOING TO GRAPH A SOLID LINE.
BECAUSE OF THE EQUAL PART,
THE LINE IS GOING TO BE PART OF THE SOLUTION.
BUT IF WE HAVE < OR >, WE'LL MAKE A DASHED LINE,
BECAUSE THE LINE WILL NOT BE PART OF THE SOLUTION.
SO TO GRAPH THIS LINE,
HOPEFULLY WE RECOGNIZE THIS AS SLOPED INTERCEPT FORM OF A LINE,
WHERE Y = MX + B FORM,
WHERE B IS THE Y INTERCEPT AND M IS THE SLOPE.
SO TO GRAPH THIS LINE, WE'LL PLOT THE Y INTERCEPT OF 4,
AND THEN FROM HERE WE'LL USE THE SLOPE
TO DETERMINE ADDITIONAL POINTS ON THIS LINE.
AND I'LL WRITE THE SLOPE AS -2/3,
WHICH REMEMBER IS THE RATIO OF THE CHANGE OF Y
TO THE CHANGE OF X.
SO FROM THE Y INTERCEPT WE'LL GO DOWN TWO UNITS
AND RIGHT THREE UNITS.
SO DOWN TWO, RIGHT THREE, DOWN TWO, RIGHT THREE, AND SO ON.
SO OUR BORDER LINE PASSES THROUGH THESE THREE POINTS.
AND AGAIN, WE ALREADY SAID IT'S GOING TO BE A SOLID LINE,
SO IT'S GOING TO LOOK SOMETHING LIKE THIS.
SO EVERY POINT ON THIS LINE IS GOING TO SATISFY
THIS INEQUALITY,
BUT ALSO EVERY POINT ON ONE SIDE OF THE LINE
IS ALSO GOING TO SATISFY THE INEQUALITY.
AND BECAUSE IT SAYS Y IS GREATER THAN OR EQUAL TO -2X + 3,
IF Y GREATER THAN OR EQUAL TO,
AND WE KNOW Y GETS GREATER AS WE MOVE UPWARD ALONG THE Y AXIS,
WE'RE TO SHADE ABOVE THE LINE.
SO EVERY SINGLE POINT IN THE HALF PLANE ABOVE THIS LINE
IS PART OF THE SOLUTION TO THIS INEQUALITY,
AND THEREFORE PART OF OUR GRAPH.
LET'S GO AHEAD AND JUST VERIFY THIS
BY CHECKING A COUPLE OF POINTS,
ONE POINT IN THE SHADED REGION,
AND ONE POINT IN THE REGION THAT'S NOT SHADED.
FOR EXAMPLE, LET'S GO AHEAD AND TEST THE POINT (3,3),
WHICH WOULD BE THIS POINT HERE,
WHICH WE'RE SAYING SHOULD BE A SOLUTION.
LET'S ALSO TEST THE ORIGIN HERE,
WHICH WE'RE SAYING IS NOT GOING TO SATISFY THE INEQUALITY.
OKAY, SO IF WE SUBSTITUTE 3 FOR X AND 3 FOR Y,
WE WOULD HAVE 3 GREATER THAN OR EQUAL TO -2/3 x 3,
WHICH I'LL WRITE AS 3/1 + 4.
THIS SIMPLIFIES.
SO WE HAVE 3 IS GREATER THAN OR EQUAL TO--
THIS WOULD BE -2 + 4 = 2. THAT'S TRUE.
AND LET'S ALSO TEST TO MAKE SURE (0,0) DOES NOT SATISFY
THIS INEQUALITY.
SO WE WOULD HAVE 0 IS GREATER THAN OR EQUAL TO -2/3 x 0
THAT'S GOING TO BE 0, + 4.
SO WE HAVE ZERO IS GREATER THAN OR EQUAL TO 4,
WHICH IS NOT TRUE, OR IT'S FALSE.
SO THIS VERIFIES THAT WE DID SHADE
THE CORRECT SIDE OF THE LINE.
AND SOME TEXTBOOKS DO RECOMMEND
THAT YOU ALWAYS PICK A TEST POINT
TO DETERMINE WHICH SIDE OF THE LINE TO SHADE ON,
MEANING IF WE TESTED THE ORIGIN, AND IT WAS FALSE THIS SIDE HERE,
WE WOULD SHADE THE OTHER SIDE.
AND IF WE TEST A POINT THAT IS TRUE,
LIKE THIS POINT HERE,
THEN WE SHADE THE SAME SIDE OF THE LINE THAT WE'RE ON,
OR THIS REGION HERE.
BUT IF THE INEQUALITY IS SOLVED FOR Y,
IT'S MUCH EASIER JUST TO ASK WHETHER YOU SHOULD SHADE
ABOVE OR BELOW THE LINE BASED UPON THE INEQUALITY.
WE'LL TAKE A LOOK AT SOME MORE EXAMPLES IN THE NEXT VIDEO.