Question - Solving Absolute Value Inequalities
Solution:
The equation given in the image is an inequality involving an absolute value:\[ \left| \frac{x + 3}{2} \right| \geq 4 \]To solve this inequality, we must consider two cases because the absolute value expression can be either positive or negative:**Case 1: The expression inside the absolute value is positive or zero**\[ \frac{x + 3}{2} \geq 4 \]To solve this, multiply both sides by 2 to get rid of the denominator:\[ x + 3 \geq 8 \]Now, subtract 3 from both sides:\[ x \geq 5 \]**Case 2: The expression inside the absolute value is negative**\[ -\left( \frac{x + 3}{2} \right) \geq 4 \]Again, multiply both sides by 2:\[ -(x + 3) \geq 8 \]Distribute the negative sign:\[ -x - 3 \geq 8 \]Now, add 3 to both sides:\[ -x \geq 11 \]Finally, multiply both sides by -1 and remember to flip the inequality sign when multiplying or dividing by a negative number:\[ x \leq -11 \]Combining both cases gives us the solution set:\[ x \leq -11 \quad \text{or} \quad x \geq 5 \]These are the values of x that satisfy the original inequality.
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