Question

(05.04 MC)

|1/4x-2|-3>4 (Underlined inequality sign)

Part A: Solve the inequality, showing all necessary steps.

Part B: Describe the graph of the solution.

HELP MEEEE

Answer 1

The solution to
`\(| (1)/(4)x - 2| - 3 > 4\)` is x < -20 or x > 36. The graph consists of two unconnected intervals on the number line.

Part A: Solve the Inequality

`\[ | (1)/(4)x - 2| - 3 > 4 \]`

1. Isolate the Absolute Value Term:

`\[ | (1)/(4)x - 2| > 7 \]`

2. Set up Two Cases:

`\[ (1)/(4)x - 2 > 7 \quad \text{or} \quad -((1)/(4)x - 2) > 7 \]`

3. Solve Each Case Separately:

- Case 1:
`\( (1)/(4)x - 2 > 7 \)`

`\[ (1)/(4)x > 9 \]`

x > 36

- Case 2:
`\( -((1)/(4)x - 2) > 7 \)`

`\[ -(1)/(4)x + 2 > 7 \]`

`\[ -(1)/(4)x > 5 \]`

x < -20

4. Combine the Solutions:

`\[ x < -20 \text{ or } x > 36 \]`

Part B: Describe the Graph of the Solution

The solution represents all real numbers except those between -20 and 36, forming two unconnected intervals on the number line.