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Express the set -3 x-6 > 3 x+12 using interval notation

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2 Answers

Hi, Treena.

The problem:  -3x -6 > 3x + 12

To solve inequalities (less than and/or greater than symbols used rather than equal sign), use the same process as solving equations (equal sign).  You want the variables to be on one side of the symbol, and the constants (plain numbers) on the other.

I tell students to look at the variables first and "mess with" the smaller one.  
In your case negative 3x is smaller than positive 3x, so I will "mess with" the negative 3x accordingly:

  -3x -6 > 3x + 12
 +3x       +3x                     .
       -6 > 6x + 12

Then work with the twelve so that the x-term is by itself:

   -6 > 6x + 12
  -12         - 12                        .
  -18 > 6x

Now divide both sides by 6 to get x by itself:

  -18 > 6x
    6      6     

  -3 > x

Now the issue is that it's written "backwards."  So flip the whole thing around so that x is in front, and the negative three is at the end.  Be sure to turn the symbol around too:

  x < -3

The reason we can do this is best demonstrated by thinking about everyday comparisons.  For example, if the size of Mercury is smaller than the size of Pluto, doesn't it imply that Pluto must be bigger than Mercury?

Anyway, interval notation is easiest explained when you look at your answer on a number line.  If you were to graph this answer on a number line, you would have a hollow circle on the neg 3 and a line extended from neg 3 to the left ("to negative infinity and beyond").

An interval indicates the lowest value to the highest value, and it's written similarly to ordered pairs.  Our lowest value cannot be read (because it's negative infinity) so we write an open parenthesis with a negative sign and an infinity sign ∞.  Our highest value is neg 3, so after writing a comma, write neg 3 followed by a close parenthesis:

(-∞, -3) <-- Interval notation for this problem.

*Note - if the problem had used a ≥ symbol where the neg three would have been included in our answer, we would need to use a bracket:  (-∞, -3]

 

I normally would not "give" the answer, but I wanted to make sure you understood how interval notation works.  Good luck on the rest of the assignment!

 

Let's first solve the inequality:

Subtract 3X from the LHS and add 6 to the RHS to get 

-6X > 18

Remember that dividing by a negative coefficient flips the inequality sign to give:

X < -3

This means, the solutions of the inequality are any values of X less than negative 3, but does not include negative 3 since the inequality was not "less than or equal to." Any value less than negative 3 is really that, all the way to negative infinity. In interval notation:

(-inf., -3) Answer

Which reads, all values of X from negative infinity to negative 3, non-inclusive.

Were the 3 included in the solution from working with a "less than or equal to," we use square brackets to indicate its inclusion. The notation would then have been:

(-inf., -3]

Notice that infinity can never be "inclusive" so never has a square bracket.