Answer:
The inequality represented by the number line is option D: 2|x-1| ≤ 10.
Here's how you can understand and solve this inequality step by step:
1. Start by breaking down the inequality into two parts: 2|x-1| and 10.
2. The absolute value sign, "| |", means that the expression inside can be either positive or negative. So, we need to consider two scenarios: (i) x-1 is positive, and (ii) x-1 is negative.
3. For scenario (i), when x-1 is positive, the inequality becomes 2(x-1) ≤ 10.
4. Simplify the inequality: 2x - 2 ≤ 10.
5. Add 2 to both sides of the inequality to isolate the variable: 2x ≤ 12.
6. Divide both sides by 2 to solve for x: x ≤ 6.
7. For scenario (ii), when x-1 is negative, the inequality becomes 2(-(x-1)) ≤ 10.
8. Simplify the inequality: -2x + 2 ≤ 10.
9. Subtract 2 from both sides of the inequality: -2x ≤ 8.
10. Divide both sides by -2. Since we are dividing by a negative number, the inequality sign flips: x ≥ -4.
11. Combining the results from both scenarios, we have -4 ≤ x ≤ 6.
12. This means that any value of x between -4 and 6 (including -4 and 6) will satisfy the inequality 2|x-1| ≤ 10.
To summarize, option D, 2|x-1| ≤ 10, represents the inequality shown on the number line. It states that the values of x that make the inequality true lie between -4 and 6, inclusive.
Explanation: