How do I solve this following inequality?  Solve the following inequality. `(3x-5)/(x+3)lt=2` (Write the answer in interval notation. Simplify the answer. Use integers or fractions for any numbers...

How do I solve this following inequality?

 

Solve the following inequality.

`(3x-5)/(x+3)lt=2`

(Write the answer in interval notation. Simplify the answer. Use integers or fractions for any numbers in the expression.)

Asked on by amber756

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lemjay | High School Teacher | (Level 3) Senior Educator

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`(3x-5)/(x+3)lt=2`

To start, change the inequality sign to "`=`" .Then, make the right side zero by subtracting both sides by 2.

`(3x-5)/(x+3)-2=0`

Simplify left side.

`(3x-5)/(x+3)- (2(x+3))/(x+3)=0`

`(3x-5)/(x+3)-(2x+6)/(x+3)=0`

`(x-11)/(x+3)=0`

Then, set the numerator and denominator equal to zero. And solve for x.

`x-11=0`                  and          `x+3=0`

     `x=11`                                      `x=-3`

The two values of x above are referred as critical numbers and it divides the number line into  three intervals. In each interval, assign a test value and substitute it to the original inequality equation to be able to determine if it satisfy the condition. The interval that satisfy or result to a true condition is the solution.

For interval `xlt-3` , test value is x=-4.

  `(3(-4)-5)/(-4+3)lt=2`

         `17lt=2 `  (False)

For interval `-3ltxlt11` , test value is x=0.

     `(3*0-5)/(0+3)lt=2`

         `-1 2/3lt=2`  (True)  

For interval, `xgt11` , test value is x=12

     `(3*12-5)/(12+3)lt=2`

          `2 1/15 lt= 2`  (False)

Furthermore, substitute the critical numbers to check if they are included in the solution.

 `x=-3`,     `(3(-3)-5)/(-3+3)lt=2`

                         `-14/0lt=2`    

Note that in fractions, zero denominator is not allowed. So, x=-3 is not a solution.

`x=11` ,         `(3*11-5)/(11+3)lt=2`

                            `2lt=2 `   (True)

Hence, the solution set of the inequality equation  `(3x-5)/(x+3)lt=2` is `-3ltxlt=11` . 

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