Question - Solving Absolute Value Inequalities with Two Cases
Solution:
To solve the inequality $$7 \leq |7y - 9|$$, we must consider two cases because the absolute value function $$|x|$$ is defined as $$x$$ when $$x \geq 0$$ and $$-x$$ when $$x < 0$$.**Case 1**: $$7y - 9 \geq 0$$If $$7y - 9$$ is non-negative, the inequality is simply:$$7 \leq 7y - 9$$Now, we solve for $$y$$:$$7 + 9 \leq 7y$$$$16 \leq 7y$$Divide both sides by 7:$$y \geq \frac{16}{7}$$**Case 2**: $$7y - 9 < 0$$If $$7y - 9$$ is negative, then $$|7y - 9| = -(7y - 9)$$, so the inequality becomes:$$7 \leq -(7y - 9)$$Now, we solve for $$y$$:$$7 \leq -7y + 9$$Rearrange:$$7y \leq 9 - 7$$$$7y \leq 2$$Divide both sides by 7:$$y \leq \frac{2}{7}$$Combining the results from both cases gives us the complete solution set for the inequality:$$\frac{2}{7} \geq y \geq \frac{16}{7}$$However, if you look carefully at the inequalities derived from both cases, you'll notice that there's actually no overlap between the two solution sets, since it's not possible for a number $$y$$ to be simultaneously greater than or equal to approximately $$2.29$$ and less than or equal to approximately $$0.29$$. This suggests we need to correct the combined inequality to reflect the actual solution.The correct combined solution for the inequality considering both cases would be:$$y \leq \frac{2}{7}$$ or $$y \geq \frac{16}{7}$$Thus, the solution set for the inequality $$7 \leq |7y - 9|$$ is that $$y$$ can be in the interval $$(-\infty, \frac{2}{7}]$$ or $$[\frac{16}{7}, +\infty)$$.
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