- WELCOME TO A LESSON ON SOLVING ONE-STEP LINEAR INEQUALITIES
IN ONE VARIABLE.
THE GOALS ARE TO SOLVE ONE-STEP LINEAR INEQUALITIES
AND ALSO TO EXPRESS THE SOLUTIONS AS AN INEQUALITY,
A GRAPH AND USING INTERVAL NOTATION.
AND TO REVIEW, HERE ARE THE THREE WAYS
THAT WE'LL EXPRESS OUR SOLUTIONS.
ON THE LEFT, WE HAVE AN INTERVAL EXPRESSED AS AN INEQUALITY.
IN THE MIDDLE, WE HAVE THE GRAPH OF THE INTERVAL.
AND THEN ON THE RIGHT, WE HAVE THE INTERVAL EXPRESSED
USING INTERVAL NOTATION.
NOTICE FOR X LESS THAN 2, WE HAVE AN OPEN POINT
ON 2 AND THEN WE SHADE TO THE LEFT AND,
AS WE MOVE TO THE LEFT ON THE NUMBER LINE,
WE WOULD APPROACH NEGATIVE INFINITY.
AND NOTICE HOW THIS IS WHY WE HAVE A NEGATIVE INFINITY
AND +2 USING INTERVAL NOTATION.
WE ALWAYS USE A ROUNDED PARENTHESIS ON POSITIVE
OR NEGATIVE INFINITY AND THEN WE USE
A ROUNDED PARENTHESIS ON 2 BECAUSE IT'S AN OPEN POINT.
NOTICE FOR X THAT IS LESS THAN OR EQUAL TO +3,
WE HAVE A CLOSED POINT ON 3, THEN WE USE A SQUARE BRACKET
ON 3 TO SHOW THAT VALUE IS INCLUDED IN THE INTERVAL.
FOR X GREATER THAN ZERO, OPEN POINT ON ZERO,
ARROW TO THE RIGHT, AND THEN INTERVAL NOTATION
IS FROM ZERO TO INFINITY, BOTH WITH ROUNDED PARENTHESES.
AND FOR X IS GREATER THAN OR EQUAL TO -4,
CLOSED POINT ON -4 AND THEN SHADE TO THE RIGHT
AND THEN, USING INTERVAL NOTATION,
WE HAVE A SQUARE BRACKET ON -4, AGAIN, SHOWING THAT IT INCLUDES
THE IN POINT AND THEN INFINITY.
NOW THE STEPS FOR SOLVING AN INEQUALITY
ARE EXACTLY THE SAME AS SOLVING LINEAR EQUATIONS
EXCEPT IF WE MULTIPLY OR DIVIDE BY A NEGATIVE
WHEN SOLVING FOR THE VARIABLE, WE MUST REVERSE
THE INEQUALITY SYMBOL.
NOW LET'S TRY TO JUSTIFY THIS EXTRA RULE.
HERE WE HAVE -2 IS LESS THAN +4.
WE KNOW THIS IS TRUE BECAUSE NEGATIVE NUMBERS
ARE ALWAYS LESS THAN POSITIVE NUMBERS.
SO IF WE DIVIDED BOTH SIDES OF THIS INEQUALITY BY,
LET'S SAY, -2, NOTICE HOW IT WOULD GIVE US THE INEQUALITY
+1 IS LESS THAN -2.