# Ex: Solve a Compound Inequality Involving OR (Union)

On January 19, 2020 by Raul Dinwiddie

– WE WANT TO SOLVE

THE COMPOUND INEQUALITY AND THEN EXPRESS THE SOLUTION

USING INTERVAL NOTATION. A COMPOUND INEQUALITY CONSISTS

OF TWO INEQUALITIES CONNECTED BY THE WORD “OR,” WHICH MEANS

THE SOLUTION WILL CONSIST OF ALL OF THE VALUES THAT

SATISFY THIS INEQUALITY AS WELL AS ALL THE VALUES THAT

SATISFY THE SECOND INEQUALITY. SO WE’RE GOING TO SOLVE

AND GRAPH BOTH OF THESE AND THEN GRAPH BOTH SOLUTIONS ON

THE SAME NUMBER LINE TO INTERPRET THE RESULTS. SO WE’LL FIRST SOLVE

THE INEQUALITY 2X – 10 IS LESS THAN -14 AND THEN WE’LL

SOLVE THE INEQUALITY -3X + 2 IS LESS THAN -10. SO TO ISOLATE X

ON THE LEFT SIDE HERE. WE’LL START BY ADDING 10 TO BOTH

SIDES OF INEQUALITY. THIS SHOULD BE 0 SO WE HAVE 2X

IS LESS THAN, THIS WOULD BE -4 AND NOW WE’LL

DIVIDE BOTH SIDES BY 2. NOW WE’RE DIVIDING BY A POSITIVE

SO WE DO NOT REVERSE INEQUALITY. IT’S ONLY WHEN MULTIPLYING

AND DIVIDING BY A NEGATIVE THAT WE REVERSE THE INEQUALITY

SYMBOL. -4 DIVIDED BY +2 IS -2. SO NOW OUR GRAPH X IS LESS THAN

-2 ON THE NUMBER LINE. AGAIN SINCE -2 DOES NOT SATISFY

THIS INEQUALITY, IT’S NOT IN THE INTERVAL, SO WE

MAKE AN OPEN POINT AT -2. THEN FOR NUMBERS LESS THAN -2

WE HAVE A ARROW TO THE LEFT. NOW I’LL GO AHEAD AND SOLVE

THE SECOND INEQUALITY. SO LET’S START BY SUBTRACTING 2

ON BOTH SIDES OF INEQUALITY. THIS WOULD BE 0 SO WE HAVE -3X

IS LESS THAN, THIS WOULD BE -12. NOW WE’LL DIVIDE BOTH SIDES

BY -3. REMEMBER WHEN WE DIVIDE

BY A NEGATIVE WE DO HAVE TO REVERSE THIS

INEQUALITY SYMBOL. SO HERE WE HAVE X AND IT’S GOING

TO BE GREATER THAN AND -12 DIVIDED BY -3 IS +4. NOW LET’S GO AHEAD AND GRAPH X

IS GREATER THAN 4. AGAIN 4 DOES NOT SATISFY THIS

INEQUALITY SO IT’S NOT IN THE INTERVAL. SO WE HAVE AN OPEN POINT

ON 4 AND THEN FOR NUMBERS GREATER

THAN 4, WE HAVE AN ARROW TO THE RIGHT. AND BECAUSE WE HAVE AN OR, WHICH MEANS UNION, THE SOLUTION

TO THIS INEQUALITY WOULD CONSIST OF INTERVAL ON THE LEFT AS WELL

AS THE INTERVAL ON THE RIGHT. SO THE SOLUTION WOULD BE X

IS LESS THAN -2 OR X IS GREATER THAN 4. NOW TO EXPRESS THIS INTERVAL

USING INTERVAL NOTATION, WE NEED TO REMEMBER

THAT AS WE MOVE TO THE RIGHT ON THE NUMBER LINE WE WOULD APPROACH POSITIVE

INFINITY AND IF WE MOVE TO THE LEFT

OF THE NUMBER LINE WE APPROACH NEGATIVE INFINITY. SO NOTICE HOW WE HAVE TWO

INTERVALS. THE FIRST INTERVAL HERE WILL

HAVE NEGATIVE INFINITY AND -2. IT’S OPEN ON -2 SO WE HAVE A

ROUNDED PARENTHESES HERE AS WELL AS ON NEGATIVE INFINITY OR WE HAVE THE INTERVAL FROM 4

TO INFINITY. IT’S OPEN ON 4 MEANING IT DOES

NOT INCLUDE 4. SO WE HAVE A ROUNDED PARENTHESES

HERE AS WELL AS TO THE RIGHT OF

INFINITY. AND SOMETIMES INSTEAD OF USING

THE WORD “OR,” YOU’LL SEE TEXTBOOKS

USE “U” FOR UNION WHICH LOOKS LIKE THIS WHICH

WOULD BE THE SAME THING AS USING OR. I HOPE THIS EXPLANATION HELPS.

Thank you! always forget when to use the dang U

What if the Union is upside down?

I like that he used terms necessary to solve the inequalities. I also like that he graphed in an alternative method and use (+/-) infinity.

This stuff is so stupid