how do you solve and graph this inequality?

Question

can someone please show me how to solve these inequality and then graph the solutions on a number line?

 

6 - x > 15

 

8x < 1 or x - 9 > -5

 

| x + 3 | > 4

              _<-- this goes under the sign before the 4

 

Please help! thanks

Answer 1

Hi Brianna,

 

We'll consider the solution in turn for each of these inequalities.

 

Solution to Inequality #1:

 

6 - x > 15

    -x > 15 - 6

    -x > 9

-x/-1 < 9/-1 (divide by -1; reverse sign of inequality)  

     x < -9 

 

Solution to Inequality #2:

 

8x < 1

  x < 1/8

 

OR

 

x - 9 > -5

     x > -5 + 9

     x > 4

 

Solution to  Inequality #3:

 

|x + 3| >= 4

 

Positive Case:

 

x + 3 >= 4

      x >= 4 - 3

      x >= 1 

 

Negative Case:

 

-1(x + 3) >= 4

     -x - 3 > = 4

          -x >= 4 + 3

          -x >= 7

-x/-1 <= 7/-1 (divide by -1; reverse sign of inequality)

     x <= -7

 

Below is the clickable link to file containing the graphs of the inequalities: 

 

https://dl.dropbox.com/s/6lxlxillh17g5p9/Inequality_Graphs.png?raw=1

 

Thanks for submitting these problems and glad to help.

 

God bless, Jordan.

Answer 2

Brianna,

             6-x>15 so we are looking for any and all the numbers x that when subtracted from 6 are greater than 15. Lets try x=4 and see if that is a possible solution i.e. 6-4=2 and 2 is not greater than 15 so x=2 is not a possible solution. We could continue to look for solutions this way but it's easier to subtract 6 from both sides of 6-x>15 which is -x>-6+15 or -x>9 now we multiply both sides by -1 which gives x<-9 so any number to the left of -9 on the real number line is a solution.

the solution set is (-∞,-9)

         You can solve the next one the same way except there are two parts x<1/8 or x>4. The solution on the real line is (-∞, 1/8) or (4,∞)

 

The next one is again done in two parts because of the absolute value function Ιx+3Ι which is really two lines: the first one is x+3 for x>-3 and -x+3 for x<-3 this graph is a "V" with the vertex at x=-3. Now you have to find out where these two lines intersect the line y=4  so x+3=4 at x=1 and -x+3=4 at x=-7 so the solution is [1,∞) and (-∞,-7]

 

It's a lot easier to solve these kind of problems by graphing the functions.

 

Hope this helps

 

Jim