Question - Solving Absolute Value Inequalities
Solution:
The inequality presented here is $$7 \leq |7y - 9|$$. To solve this, we need to consider the absolute value |7y - 9| and split the inequality into two cases because the expression inside the absolute value can be either positive or negative.The absolute value inequality says that the expression inside the absolute value is either greater than or equal to 7 or less than or equal to -7.First, we'll handle the case where the expression inside the absolute value is non-negative:1. $$7y - 9 \geq 7$$2. $$7y \geq 16$$3. $$y \geq \frac{16}{7}$$Next, we handle the case where the expression inside the absolute value is non-positive:1. $$7y - 9 \leq -7$$2. $$7y \leq 2$$3. $$y \leq \frac{2}{7}$$Combining both inequalities, we get the solution set for y:$$y \leq \frac{2}{7}$$ or $$y \geq \frac{16}{7}$$.Graphically, this means y is either in the interval $$(-\infty, \frac{2}{7}]$$ or in the interval $$[\frac{16}{7}, \infty)$$.
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