How do you graph and solve # abs[2-x/2]-1<=1#?

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To graph and solve the inequality ( |2 - \frac{x}{2}| - 1 \leq 1 ), you can follow these steps:

1. Isolate the Absolute Value Expression: Add 1 to both sides of the inequality to isolate the absolute value expression.

   ( |2 - \frac{x}{2}| \leq 2 )

2. Break into Two Cases: Since we have an absolute value expression, we need to consider two cases: ( 2 - \frac{x}{2} \geq 0 ) and ( 2 - \frac{x}{2} < 0 ).

   • Case 1: ( 2 - \frac{x}{2} \geq 0 )
     Solve ( 2 - \frac{x}{2} \geq 0 ) for ( x ).

   • Case 2: ( 2 - \frac{x}{2} < 0 )
     Solve ( -(2 - \frac{x}{2}) \leq 0 ) for ( x ).

3. Graph the Solutions: Once you've found the solutions for each case, graph them on a number line. The solutions will be the intervals where the inequality holds true.

4. Check for Equality: Remember to check if equality holds for any endpoints of the intervals.

5. Combine the Solutions: Combine the solutions from both cases to get the overall solution set for the inequality.

6. Graph the Final Solution: Once you have the solution set, graph it on a number line to represent all values of ( x ) that satisfy the original inequality.

Following these steps, you can graph and solve the inequality ( |2 - \frac{x}{2}| - 1 \leq 1 ).