# Solve the inequality 10 l 10n - 8 l < 80

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Given the inequality 10 l 10n - 8 l < 80.

We need to find all possible values of n.

First, we will solve the same way we solve the equality.

Let us divide by 10.

==> l 10n -8 l < 8

Now we have two cases.

Case (1):

==> (10n-8) < 8

Now add 8 to both sides.

==> 10n < 16.

Now we will divide by 10.

==> n < 1.6

**Then the values of n belongs to the interval ( -inf, 1.6)**

Case (2):

==> -(10n-8) < 8

==> -10n +8 < 8

Now subtract 8 from both sides.

==> -10n < 0

Now we will divide by -10 and reverse the inequality.

==> n > 0

**Then n values belong to the interval ( 0, inf)**

**==> n = ( -inf, 1.6) U ( 0, inf)**

The first step is to simplify the given expression, dividing it by 10 both sides:

10|10n - 8|<80

|10n - 8|<8

Now, we'll discuss the absolute value of the expression 10n - 8:

Case 1) 10n - 8 for 10n - 8>=0

10n>=8

n>=8/10

n>=4/5

We'll solve the inequality:

10n - 8 < 8

10n < 16

n < 16/10

n < 1.6

**The interval of admissisble value of n, for this situation, is [4/5 , 1.6).**

Case 2) 8 - 10, for 10n - 8<0

n<4/5

We'll solve the inequality:

8 - 10n < 8

-10n < 0

10n > 0

n > 0

**The interval of admissible values for n, for this case, is (0 , 4/5).**