SOLUTION: Solve the inequality. Express your answer using set notation and interval notation. Graph the solution set. 0 < [1 - (x/3)] < 1

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Question 1208858: Solve the inequality. Express your answer using set notation and interval notation. Graph the solution set.


0 < [1 - (x/3)] < 1


Found 4 solutions by timofer, greenestamps, math_tutor2020, Edwin McCravy:
Answer by timofer(105)   (Show Source): You can put this solution on YOUR website!
Taking those square brackets as just grouping symbols and not as indication for absolute value










<------------------------()================()------------->
                         0                  3

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!




Multiply by 3 to clear fractions:


Subtract 3 to get the "-x" by itself in the middle:



Multiply by -1 to leave just "x" in the middle. Remember that multiplying by a negative reverses the direction of the inequality.



Rewrite the inequality in the standard form, with the smaller number on the left.

ANSWER: or (0,3)
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

There are at least two methods we can use to solve.

Method 1

0 < 1 - (x/3) < 1
0-1 < 1 - (x/3)-1 < 1-1 ........ see note1
-1 < -x/3 < 0
-1*(-3) > -3*(-x/3) > -3*0 ........ see note2
3 > x > 0
0 < x < 3

Notes:
  1. Subtract 1 from all sides
  2. Multiply all sides by -3. Multiplying all sides by a negative number will flip the inequality sign.
Anyways, we conclude the solution is 0 < x < 3

The interval notation would be (0,3)
Be sure not to mix this up with ordered pair notation.

The graph on a number line will have open holes at 0 and 3; with shading in between.
In words: x is any number between 0 and 3 excluding each endpoint.

----------------------------------------------------------------------------------------

Method 2

a < b < c is the same as saying a < b and b < c

0 < 1 - (x/3) < 1 breaks down into 0 < 1 - (x/3) and 1 - (x/3) < 1

Let's solve the first part.
0 < 1 - (x/3)
x/3 < 1
x < 3

Now solve the second part.
1 - (x/3) < 1
-x/3 < 0
x > 0 .... see note2 mentioned earlier

We conclude that x > 0 and x < 3
Put another way: 0 < x and x < 3
Those collapse or glue together to get 0 < x < 3

Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!

Set notation:  {x | 0 < x < 3}

Interval notation:  (0,3)

[It is unfortunate that " (0,3) " can be used either for an 
interval or a point. But we have to live with it, and go by context.] 

Edwin
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