Question

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

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asked 2021-06-04

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

Answers (1)

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,
\(-3<\frac{2}{x}-4\)
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 \((\frac{2}{7},2)\)

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