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Answer:

To solve the inequality [tex]||x - 2| - x + 3| < |x|[/tex], we can break it down into two cases based on the signs of [tex](x - 2)[/tex] and [tex](x - 3)[/tex].

Case 1: [tex](x - 2)[/tex] and [tex](x - 3)[/tex] are both non-negative (greater than or equal to 0). In this case, the absolute value expressions simplify to:

• [tex]|x - 2| = (x - 2)[/tex]
• [tex]|x - 3| = (x - 3)[/tex]

So, our inequality becomes:

• [tex](x - 2) - x + 3 < x[/tex]

Simplifying this expression, we have:

• [tex]x - 2 - x + 3 < x[/tex]
• [tex]1 < x[/tex]

Case 2: [tex](x - 2)[/tex] and [tex](x - 3)[/tex] are both negative (less than 0). In this case, the absolute value expressions simplify to:

• [tex]|x - 2| = -(x - 2) = -x + 2[/tex]
• [tex]|x - 3| = -(x - 3) = -x + 3[/tex]

So, our inequality becomes:

• [tex](-x + 2) - x + 3 < x[/tex]

Simplifying this expression, we have:

• [tex]-2x + 5 < x[/tex]
• [tex]5 < 3x[/tex]
• [tex]\frac{5}{3} < x[/tex]

Therefore, the solution set for the inequality is [tex]\left(\frac{5}{3}, \infty\right)[/tex].

• Hence, the correct answer is (a) [tex]\left(\frac{5}{3}, \infty\right)[/tex].

Step-by-step explanation:

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