Part A: Solve the inequality, showing all necessary steps.
To solve the inequality, let's break it down into two cases: when the quantity inside the absolute value is positive and when it is negative.
Case 1: When the quantity inside the absolute value is positive:
1/4(x - 2) - 3 ≥ 4
To solve this inequality, let's first distribute the 1/4:
1/4x - 1/2 - 3 ≥ 4
Next, combine like terms:
1/4x - 7/2 ≥ 4
To isolate x, let's add 7/2 to both sides:
1/4x ≥ 4 + 7/2
Combining the terms on the right side:
1/4x ≥ 8/2 + 7/2
1/4x ≥ 15/2
To get rid of the fraction, let's multiply both sides by 4:
4 * (1/4x) ≥ 4 * (15/2)
Simplifying:
x ≥ 30/2
x ≥ 15
Case 2: When the quantity inside the absolute value is negative:
-(1/4(x - 2)) - 3 ≥ 4
To solve this inequality, let's first distribute the negative sign:
-1/4x + 1/2 - 3 ≥ 4
Next, combine like terms:
-1/4x - 5/2 ≥ 4
To isolate x, let's add 5/2 to both sides:
-1/4x ≥ 4 + 5/2
Combining the terms on the right side:
-1/4x ≥ 8/2 + 5/2
-1/4x ≥ 13/2
To get rid of the fraction, let's multiply both sides by -4 (remember to reverse the inequality because we're multiplying by a negative number):
-4 * (-1/4x) ≤ -4 * (13/2)
Simplifying:
x ≤ -26/2
x ≤ -13
Combining the solutions from both cases:
x ≥ 15 or x ≤ -13
Part B: Describe the graph of the solution.
The solution to the inequality x ≥ 15 or x ≤ -13 represents all the values of x that make the absolute value expression greater than or equal to 4.
On the number line, we can represent the solution graphically. We would have a shaded region to the right of 15 (including 15) and a shaded region to the left of -13 (including -13). This indicates all the possible values of x that satisfy the inequality. The shaded regions will be separated by an open circle at -13 and 15 since the inequality is "greater than or equal to" but not "equal to."