Question

Select the correct answer. What is the solution to |x − 4| ≥ 7? A. -3 ≤ x ≤ 11 B. -11 ≤ x ≤ 3 C. x ≥ 11 or x ≤ -3 D. x ≥ 3 or x ≤ -11

Answer 1

x ≥ 11 or x ≤ -3

Step-by-step explanation:

Hi there! Please take a look at the following explanation and let me know in comment if you have any questions!

Given:

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To solve an absolute inequality, consider the following theorems.

Theorems:

For b ≥ 0:

(1) If |x-a| ≥ b then we have x-a ≥ b or x-a ≤ -b so that we end up with x ≥ b+a, x ≤ -b+a

(2) If |x-a| ≤ b then we have -b ≤ x-a ≤ b which can be simplified to -b+a ≤ x ≤ b+a

Step:

From the inequality, we use the (1) theorem.

`\displaystyle \large{x-4\geq 7, x-4\leq -7}\\\\\displaystyle \large{x\geq 7+4, x\leq -7+4}\\\\\displaystyle \large{x\geq 11, x \leq -3}`

Therefore, the answer is x ≥ 11 or x ≤ -3

Answer 2

C) x ≥ 11 or x ≤ -3

Step-by-step explanation:

Given following: |x − 4| ≥ 7

Apply absolute rule: If |u| ≥ 0 then u ≤ -a, u ≥ a

Solving stepwise:

⇒ x - 4 ≤ -7, x - 4 ≥ 7

change sides

⇒ x ≤ -7 + 4, x ≥ 7 + 4

add/subtract integers

⇒ x ≤ -3, x ≥ 11