Help with a system of inequalities with absolute values { <mtable rowspacing="4pt" c

hornejada1c

hornejada1c

Answered question

2022-07-10

Help with a system of inequalities with absolute values
{ | x 3 | < 2 x | 2 x + 5 | > 3
The steps I'm taking are:
Finding the absolute values sings, so for x 3 0 we have x 3 therefore
| x 3 | = { x 3 for  x 3 x + 3 for  x < 3
and
2 x + 5 0 we have x 2 5 therefore
| 2 x + 5 | = { 2 x + 5 for  x 2 5 2 x 5 for  x < 2 5
So I build a few systems with the complete inequalities, for the first one we have:
{ x 3 x 3 < 2 x = x > 3
So the solution here would be x > 3, then:
{ x < 3 x + 3 < 2 x = x > 1
The solution would be 1 < x < 3, then
{ x 2 5 2 x + 5 > 3 = x > 1
So the solution of the system is x > 1, then
{ x < 2 5 2 x 5 > 3 = x < 4
And the solution is x < 4
What am I doing wrong?

Answer & Explanation

Allison Pena

Allison Pena

Beginner2022-07-11Added 14 answers

Since 2 x > | x 3 | we get x > 0 so 2 x + 5 > 0 and so 2 x + 5 > 3 so x > 2 which is nothing new. So x > 0 and | x 3 | < 2 x so after squaring we get
x 2 6 x + 9 < 4 x 2 3 x 2 + 6 x 9 > 0
or
( x + 3 ) ( x 1 ) > 0 x 1 > 0
so x > 1 .
letumsnemesislh

letumsnemesislh

Beginner2022-07-12Added 6 answers

There is actually only the first inequality to be solved:
| 2 x + 5 | > 0 | x 3 | < 2 x | x 3 | + 5 > 3
So, you only need to solve | x 3 | < 2 x while x > 0:
2 x < x 3 < 2 x { 3 x > 3 x > 1 x > 3  does not extend the solution

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