Question

Solve each inequality ||2x-1|-2|>3

Answer 1

Solution: x < -2 or x > 3

Interval notation: (-∞, -2) ∪ (3, ∞)

Explanation:

Given inequality:

`||2x-1|-2| > 3`

Apply the absolute rule:

`\textsf > a$, $a > 0$ \;then \;$u > a$ \;or \;$u < -a$.`

Therefore:

`\textsf{Case 1}: \quad |2x-1|-2 > 3`

`\textsf{Case 2}: \quad |2x-1|-2 < -3`

Solve each case independently.

Case 1

Isolate the absolute value on one side of the equation:

`\begin{aligned}\implies |2x-1|-2& > 3\\\implies |2x-1| & > 5 \end{aligned}`

Apply the absolute rule:

`\textsfu.`

`\begin{aligned}\underline{\textsf{Equation 1}} & & \quad \quad\underline{\textsf{Equation 2}}\\2x-1 & > 5 & 2x-1 & < -5\\2x& > 6 & 2x& < -4\\x& > 3 & x& < -2\end{aligned}`

Therefore, x < -2 or x > 3.

Case 2

Isolate the absolute value on one side of the equation:

`\begin{aligned}\implies |2x-1|-2& < -3\\\implies |2x-1| & < -1 \end{aligned}`

As an absolute value cannot be less than zero, there is no solution for x∈R.

Solution

Solution: x < -2 or x > 3

Interval notation: (-∞, -2) ∪ (3, ∞)