Question

Write the solution set of the equation |2 − x| = x − 2.

Answer 1

\[\text{ We have }, \]
\[\left| 2 - x \right| = x - 2\]
\[\text{ Now 2 cases arise } . \]
\[\text{ CASE 1: When 2 } - x \geq 0, \text{ then } \left| 2 - x \right| = 2 - x\]
\[ \Rightarrow \left| 2 - x \right| = x - 2\]
\[ \Rightarrow 2 - x = x - 2\]
\[ \Rightarrow 2x = 4\]
\[ \Rightarrow x = 2\]
\[\text{ So, this condition is satisfied when } x = 2 . \]
\[\text{ CASE 2: When } 2 - x < 0 \left( \text{ i . e . when } x > 2 \right), \text{ then } \left| 2 - x \right| = - (2 - x)\]
\[ \Rightarrow \left| 2 - x \right| = x - 2\]
\[ \Rightarrow - (2 - x) = x - 2\]
\[ \Rightarrow - 2 + x = x - 2\]
\[ \Rightarrow - 2 = - 2\]
\[\text{ So, this condition is satisfied when } x > 2\]
\[\text{ Hence, from the given two cases, the solution set of the given equation is } [2, \infty )\]
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