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Question # Solve the absolute value and find intervals.|\frac{2}{x}-4|<3

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ANSWERED Solve the absolute value and find intervals.
$$|\frac{2}{x}-4|<3$$ 2021-06-05

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,
$$-3<\frac{2}{x}-4$$
$$\Rightarrow -3+4 < \frac{2}{x}$$
$$\Rightarrow 1 < \frac{2}{x}$$
Multiply by x on both sides,
$$\Rightarrow x<2$$
And
$$\frac{2}{x}-4<3$$
$$\Rightarrow \frac{2}{x}<7$$
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 $$(\frac{2}{7},2)$$