Absolute Inequality 2535C3

1. **State the problem:** Solve the inequality $$|10 - t| - 3 \geq 11$$ and write the solution set in interval notation if possible. 2. **Isolate the absolute value:** Add 3 to both sides: $$|10 - t| - 3 + 3 \geq 11 + 3$$ $$|10 - t| \geq 14$$ 3. **Recall the definition of absolute value inequality:** For $$|x| \geq a$$ where $$a > 0$$, the solution is $$x \leq -a$$ or $$x \geq a$$. 4. **Apply this to our inequality:** $$10 - t \leq -14 \quad \text{or} \quad 10 - t \geq 14$$ 5. **Solve each inequality separately:** - For $$10 - t \leq -14$$: $$-t \leq -14 - 10$$ $$-t \leq -24$$ Multiply both sides by $$-1$$ and reverse inequality: $$t \geq 24$$ - For $$10 - t \geq 14$$: $$-t \geq 14 - 10$$ $$-t \geq 4$$ Multiply both sides by $$-1$$ and reverse inequality: $$t \leq -4$$ 6. **Combine the solution:** $$t \leq -4 \quad \text{or} \quad t \geq 24$$ 7. **Write the solution set in interval notation:** $$(-\infty, -4] \cup [24, \infty)$$ 8. **Check if the solution set is empty:** The solution set is not empty; it includes all $$t$$ less than or equal to $$-4$$ and all $$t$$ greater than or equal to $$24$$. **Final answer:** The solution set is $$(-\infty, -4] \cup [24, \infty)$$.