# 3 l 2 - 5x l+ 1 < 10. Find all values of x which make this inequality true.I am hoping I didn't confuse anyone with how I wrote my question :)

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3 l2- 5xl + 1< 10

First we will free the terms between the absolute values sign:

==> subtract 1 from both sides:

==> 3 l 2- 5xl < 9

Now divide by 3:

==> l 2- 5x l < 3

Now we have 2 cases:

(2- 5x) < 3 OR -(2-5x) < 3

Let us solve both cases:

2- 5x < 3

==> -5x < 1

Now divide by -5 ( remeber to reverse the inequality)

**==> x > -1/5**

** **

Now for case (2):

-(2-5x) < 3

==> -2 + 5x < 3

==> 5x < 5

**==> x < 1**

**==> -1/5 < x < 1 **

**==> x belongs to the interval (-1/5, 1)**

To determine all x values, we'll work a bit over the given expression.

3 l 2 - 5x l+ 1 < 10

We'll subtract 1 both sides:

3 l 2 - 5x l < 10 - 1

3 l 2 - 5x l < 9

We'll divide by 3:

l 2 - 5x l < 9

We'll write the expression as a double inequality:

-9 < 2 - 5x < 9

We'll solve the left inequality:

-9 < 2 - 5x

-11 < -5x

We'll use the symmetric property:

-5x > -11

We'll multiply by -1:

5x < 11

We'll divide by 5:

x < 11/5

We'll solve the right inequality:

2 - 5x < 9

-5x < 9-2

-5x < 7

We'll divide by -5:

x > -7/5

**The admissible values for x belong to the interval (-7/5 , 11/5).**