Question

1. |x - 3| - 2 > -1
2. |x + 2| - 4 ≥ -2

Explanation:

Step1: Isolate the absolute value

Add 2 to both sides:
$|x-3| - 2 + 2 > -1 + 2$
$|x-3| > 1$

Step2: Split absolute value inequality

For $|A| > b$, solve $A > b$ and $A < -b$:
$x-3 > 1$ and $x-3 < -1$

Step3: Solve first inequality

Add 3 to both sides:
$x-3+3 > 1+3$
$x > 4$

Step4: Solve second inequality

Add 3 to both sides:
$x-3+3 < -1+3$
$x < 2$

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Step1: Isolate the absolute value

Add 4 to both sides:
$|x+2| - 4 + 4 \geq -2 + 4$
$|x+2| \geq 2$

Step2: Split absolute value inequality

For $|A| \geq b$, solve $A \geq b$ and $A \leq -b$:
$x+2 \geq 2$ and $x+2 \leq -2$

Step3: Solve first inequality

Subtract 2 from both sides:
$x+2-2 \geq 2-2$
$x \geq 0$

Step4: Solve second inequality

Subtract 2 from both sides:
$x+2-2 \leq -2-2$
$x \leq -4$

Answer:

For problem 4: $x < 2$ or $x > 4$
(To graph: Open circles at $x=2$ and $x=4$, shade all regions to the left of 2 and to the right of 4)

For problem 6: $x \leq -4$ or $x \geq 0$
(To graph: Closed circles at $x=-4$ and $x=0$, shade all regions to the left of -4 and to the right of 0)