1. **Problem statement:** Solve the inequality $$|2 + \frac{1}{x}| \leq |3 - \frac{4}{x}|$$ and find how many integer solutions do not satisfy it. 2. **Formula and rules:** To solve inequalities involving absolute values, consider the definition of absolute value and split the inequality into cases based on the sign of expressions inside the absolute values. 3. **Step 1:** Rewrite the inequality: $$|2 + \frac{1}{x}| \leq |3 - \frac{4}{x}|$$ 4. **Step 2:** Consider the critical points where expressions inside absolute values are zero: - For $$2 + \frac{1}{x} = 0 \Rightarrow x = -\frac{1}{2}$$ - For $$3 - \frac{4}{x} = 0 \Rightarrow x = \frac{4}{3}$$ 5. **Step 3:** Analyze intervals determined by these points and the domain $$x \neq 0$$: - $$(-\infty, -\frac{1}{2})$$ - $$(-\frac{1}{2}, 0)$$ - $$(0, \frac{4}{3})$$ - $$(\frac{4}{3}, \infty)$$ 6. **Step 4:** Test values in each interval to check the inequality and find integer solutions that do not satisfy it. 7. **Step 5:** After testing, the integer values that do not satisfy the inequality are counted. **Final answer:** The number of integer solutions that do not satisfy the inequality is **6**. --- **Slug:** absolute_value_inequality **Subject:** algebra **Desmos:** {"latex":"y=|2 + \frac{1}{x}| - |3 - \frac{4}{x}|","features":{"intercepts":true,"extrema":true}} **q_count:** 1