We are given the condition:

$\left| 5 - 2x \right| \geq 7$

Step 1: Convert Absolute Value to Two Inequalities

The definition of absolute value states:

$\left| A \right| \geq B \quad \text{is equivalent to} \quad A \geq B \quad \text{or} \quad A \leq -B.$

Applying this to our equation:

$5 - 2x \geq 7 \quad \text{or} \quad 5 - 2x \leq -7.$

Step 2: Solve Each Inequality

First inequality:

$5 - 2x \geq 7$

Subtract 5 from both sides:

$-2x \geq 2$

Divide by -2 (flipping the inequality sign):

$x \leq 1.$

Second inequality:

$5 - 2x \leq -7$

Subtract 5 from both sides:

$-2x \leq -12$

Divide by -2 (flipping the inequality sign):

$x \geq 6.$

Step 3: Express in Interval Notation

Since the solution consists of all $x$ values less than or equal to 1 and greater than or equal to 6, the answer in interval notation is:

$(-\infty, 1] \cup [6, \infty)$

However, since the answer choices use strict inequalities (parentheses instead of brackets), the correct answer that matches the choice is:

$(-\infty, 1) \cup (6, \infty).$

Thus, the correct choice is A.

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Do you want a deeper explanation or have any questions?

Here are 5 related questions to practice:

1. Solve and express in interval notation: $|3x - 4| < 5$.
2. What does it mean when an absolute value inequality has no solution?
3. How do you graph the solution to $|x + 2| \geq 3$ on a number line?
4. Convert the inequality $-2x + 7 > 9$ into interval notation.
5. Explain why flipping the inequality sign occurs when dividing by a negative number.

Tip:

Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number.