Question

Solve the following inequality ==>



$| |x - 2| - 3| \leqslant 0 \\ \: \\ \\ answer -> \\ x = 5 \: or \: x = - 1$

Answer 1

The absolute value l lx-2l -3l is less than or equal to 0.

Let take an example that lxl is less than or equal to 5.

If the value of x is less than 5 than -5≤x≤5

so -5≤x or x≤5

-( lx-2l -3)≤0 ≤lx-2l-3

⇒-lx-2l ≤ 3 ≤lx-2l [by adding 3 in all inequalities]

⇒-lx-2l ≤3 or 3≤lx-2l

⇒lx-2l ≥3 (sign changes) or 3≤lx-2l

⇒ lx-2l ≥3 (both of above means same)

⇒x-2≥ +-3

⇒x≥ (+-3) +2

⇒ x ≥5,-1

⇒x≥5 or x≥-1

Answer 2

$\textbf {Hey there, here is the solution.}$

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| |x -2| - 3 | ≤ 0

Remove first modulus bracket:

|x -2| - 3 = ±0

Since 0 has not positive or negative:

|x -2| - 3 = 0

Add 3 to both sides:

|x -2| = ±3

Remove modulus bracket:

x -2 = ±3

Add 3 to both sides:

x = 3 + 2 or x = -3 + 2

Evaluate:

x = 5 or x = -1

Answer: x = 5 or x = -1

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⭐$\textbf {Cheers}$⭐