Solve the system: | 3 x + 2 | ≥ 4 | x − 1...

Misael Matthews

Misael Matthews

Answered

2022-07-01

Solve the system:
| 3 x + 2 | 4 | x 1 |
x 2 + x 2 2 + 3 x 2 x 2 0

Answer & Explanation

Elianna Douglas

Elianna Douglas

Expert

2022-07-02Added 23 answers

If x 1 we have
| 3 x + 2 | 4 | x 1 | 3 x + 2 4 x 4 x 6.
So [ 1 , 6 ] is solution of the first inequality. Now, if 2 / 3 x 1 we have
| 3 x + 2 | 4 | x 1 | 3 x + 2 4 4 x x 2 / 7.
So [ 2 / 7 , 1 ] is solution of the system. Finally, if x 2 / 3 then
| 3 x + 2 | 4 | x 1 | 3 x 2 4 4 x x 6 ,
which is impossible. That is, the solution set of the first inequality is [ 2 / 7 , 6 ] . (The idea is to solve 3 x + 2 = 0 and x 1 = 0 and study the inequality on each region you obtain.)
Now, we will work with the second inequality. Write it as
( x 1 ) ( x + 2 ) 2 ( x 2 ) ( x + 1 / 2 ) 0.
Study the sign on ( , 2 ) ,, ( 2 , 1 / 2 ) ,, ( 1 / 2 , 1 ) ,, ( 1 / 2 , 1 ) and ( 1 , ) . You should obtain that the set solution is ( , 2 ] ( 1 / 2 , 1 ] ( 2 , ) . (The idea is to solve x 2 + x 2 = 0 and 2 + 3 x 2 x 2 = 0 and study the inequality on each region you obtain. Note that we can't divide by 0. So x 1 / 2 and x 2.
Finally, one gets the intersection to obtain ( 2 / 7 , 1 ] ( 2 , 6 ] .
April Bush

April Bush

Expert

2022-07-03Added 6 answers

Thanks, I appreciate your help!

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