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-9<-1+4x<15 and 8-x>15 or 6x-13>11

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2 Answers

Solving inequality is very like solving equation. The only difference: if divide or multiply both sides of inequality by negative number, we have to change sign of inequality to opposite.
~~~~~~~~~~

1.    8  –  x  >  15
   − 8            − 8

    − x  >  7
÷(- 1)  ÷(- 1)

    x < - 7  ,  or all real numbers out of interval (- ∞, - 7) 

 <====|=====o———|————>
- ∞   - 10       - 7        10       + ∞

Let's verify the answer:

for x = -10,

8 - (-10) > 15
8 + 10 > 15
18 > 15   (True)

for x = 10,

8 - 10 > 15
- 2 > 15   (False)

2.  6x  -  13   >   11
           + 13     + 13

     6x  >  24
  ÷ 6      ÷ 6

    x > 4 , or all real numbers out of interval (4, + ∞) 

  <---------|----o======|=====>
− ∞         0     4             10       + ∞

Check the answer:

for x = 0,
6*0 - 13 > 11
   - 13 > 11  (False

for x = 10,
6*10 - 13 > 11
60 - 13 > 11
47 > 11   (True)

3.  - 9 < - 1 + 4x < 15
    + 1    + 1          + 1

    - 8  <  4x  <  16
   ÷ 4   ÷ 4      ÷ 4

    - 2 < x < 4 , or all real numbers out of interval (-2, 4)

<-  -----|-----o====|======o-----|------>
− ∞  − 3   − 2         0             4      6    + ∞

Check the answer:

for x = - 3,
- 9 < - 1 + 4*(-3) < 15
- 9 < -13 < 15   (False)
( - 9 > - 13) !!!

for x = 0,
- 9 < - 1 + 4*0 < 15
- 9 < 3 < 15    (True)

for x = 6,
- 9 < - 1 + 4*6 < 15
- 9 < 23 < 15   (False)
 ( 23 > 15 ) !!!
Hi Jackie,
 
If I am reading your problem correctly, you have 3 separate functions. Is that accurate? If, so, here are the steps:
 
1.  Solve each function for x
     
  • For -9<-1+4x<15, you will break it down into 2 separate functions.
 
     -9<-1 + 4x                 AND     -1+4x<15  
    -9+1< -1 +1 +4x                     -1+1+4x<15+1
      -8<4x                                            4x<16
(divide each side by 4)                        (divide each side by 4)
     -2<x                                                x<16
 
 we are left with -2<x<16
 
so for this function, the x can be any number between -2 and 16. (so -1, 0, 1, 2, 3, 4, 5 etc)
 
 
  • You will need to do the same for 8-x>15
                                               -8 +8 -x>15 - 8
                                                        -x >7  (I can not have a negative value for x, so I need to switch the sign of the numbers AND the symbol between the, because I am multiplying each side by -1)
                                                         x < 7
 
 
 
  • Let's repeat the steps again for 6x-13>11
                                              6x - 13 + 13 > 11 +13
                                               6x > 24
                                             (divide each side by 6)
                                               x>4
 
I need to know if these are 3 separate functions or not, to help you graph it.