How to calculate Answer

Question

|x-2|-|3x-10|>-6

Answer 1

We should check the inequality for 3 intervals of x

(-infinity,2) , (2,10/3) & (10/3,infinity)

 

These intervals cover all the values of x and are deduced by equating the values in mod as zero.

 

1st interval  (-infinity, 2)

 

In this interval,  the values inside both the mod bracket will be less than zero so when we open the mod bracket, we should place a negative sign before it. So when we open the brackets, the eqn will look like

 

-(x-2) - (-(3x-10)) > -6

-x+2+3x-10> -6 

2x > 2

x > 1

 

So for this interval we have the solution as x ∈ (1,2)

 

2nd Interval (2,10/3)

 

In this interval, the value inside the first mod bracket  | x-2 | will be positive whereas in the second mod bracket | 3x-10| will be negative. So we need to open the first mod bracket with a positive sign and the second mod bracket with a negative sign

 

+(x-2) - (-(3x-10)) > -6

x-2+3x-10 > -6

4x > 6

x > 3/2

 

So for this interval we have the solution as x ∈ (3/2,10/3)

 

3rd Interval (10/3,infinity)

 

In this interval, the values inside both the mod bracket will be more than zero so when we open the mod bracket, we should place a positive sign before it. So when we open the brackets, the eqn will look like

 

+(x-2)-(+(3x-10)) > -6

x-2-3x+10 > -6

-2x > -14

x < 7

 

So for this interval we have the solution as x ∈ (10/3,7)

 

Now after combining the solutions for all the three intervals, our final solution will be

 

x ∈ (1,2)∪(3/2,10/3)∪(10/3,7)

x ∈ (1,7)