# find x values if l 2x -5 l + 7 < 10

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### 2 Answers

l 2x -5 l + 7 < 10

First lest us subtract 7 from both sides:

==> l 2x - 5 l < 10 - 7

==> l 2x -5 l < 3

Now we have two cases:

Case(1);

( 2x -5) < 3

Add 5 to both sides.

==> 2x < 8.

Now we will divide both sides by 2.

==> x < 4

**==> x = ( -inf , 4)**

case(2):

-(2x-5) < 3

==> -2x + 5 < 3.

Subtract 5 from both sides.

==> -2x < -2.

Now we will divide by -2 and reverse the inequality.

==> x > 1

==> x = ( 1, inf)

Then the solution is:

**x = ( -inf, 4) U ( 1, inf)**

To find values of the inequality |2x-5| < 10 .

We know that |x-n| = x-n ix x >n and |x-n| = n-x if x< n.

Therefore |2x-5| = 2x-5 for 2x-5 > 0. Or x > 5/2.

Therefore or x > 5/2 , |2x-5| < 10 implies 2x-5 < 10

2x-5 < 10 implies 2x-5+5 < 10+5 , as we added equals to both side.

2x < 15.

2x < 15 implies x < 15/2 = 7.5 , as we divided both sides by positive equals.

Therefore **x>2.5 , x < 7.5.**

When x < 2.5 , |2x-5| < 10 implies 5-2x < 10.

We add 2x to both sides:

5 < 10 +2x.

5-10 < 10+2x-10.

-5 < 2x

-5/2 < 2x/2

-2.5 < x. Or x > -2.5.

Therefore ** x < -2.5 for any x < =2.5.**

**Therefore x is in the interval (-2.5 , 7.5} but the strict inequality does allow x to take boundary values.**