Solve the absolute value and find intervals. |2x−4|&lt;3

Step 1 Given : |2x−4|&lt;3 To find : Solve the absolute value Find the intervals Step 2 Consider, |2x−4|&lt;3 Use the property of the absolute value function : If |x|0 then -a Here, Substitute x=2x−4 and a=3 Therefore, |2x−4|&lt;3 ⇒−3&lt;2x−4&lt;3 Step 3 Consider, −3&lt;2x−4 Add 4 on both sides, ⇒−3+4&lt;2x ⇒1&lt;2x Multiply by x on both sides, ⇒x&lt;2 And 2x−4&lt;3 Add 4 on both sides, ⇒2x&lt;7 Multiply by x on both sides, ⇒2&lt;7x Divide by 7 on both sides, ⇒27 Therefore, we get x&lt;2 and 27&lt;x which gives that 27 Step 4 Now we have 27: Hence, the interval is (27,2)

Expert Answer to "Solve the absolute value and find intervals. |2x−4|<3"

facas9

Answered question

2021-06-04

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

Answer & Explanation

comentezq

Skilled2021-06-05Added 106 answers

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,
$\Rightarrow 2 < 7 x$
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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