Step 1

Given :

\(|\frac{2}{x}-4|<3\)

To find :

Solve the absolute value

Find the intervals

Step 2

Consider,

\(|\frac{2}{x}-4|<3\)

Use the property of the absolute value function :

If \(|x|0\) then -a Here, Substitute \(x =\frac{2}{x}-4\) and \(a = 3\)

Therefore,

\(|\frac{2}{x}-4|<3\)

\(\Rightarrow -3 < \frac{2}{x}-4<3\)

Step 3

Consider,

\(-3<\frac{2}{x}-4\)

Add 4 on both sides,

\(\Rightarrow -3+4 < \frac{2}{x}\)

\(\Rightarrow 1 < \frac{2}{x}\)

Multiply by x on both sides,

\(\Rightarrow x<2\)

And

\(\frac{2}{x}-4<3\)

Add 4 on both sides,

\(\Rightarrow \frac{2}{x}<7\)

Multiply by x on both sides,

\(\Rightarrow 2 <7x\)

Divide by 7 on both sides,

\(\Rightarrow \frac{2}{7}\)

Therefore, we get \(x<2\) and \(\frac{2}{7} < x\) which gives that \(\frac{2}{7}\)

Step 4

Now we have \(\frac{2}{7}:\)

Hence, the interval is \((\frac{2}{7},2)\)