Twice a number, x, is added to 5, the result is at least 11. What is the range of values of x?

Analyze the question

"Twice a number, x, is added to 5" gives 2x + 5. "The result is at least 11" means the result is greater than or equal to 11.

Concept

"At least" means the smallest allowed value is that number, so we use ≥. (Compare: "at most" means ≤.)

Solution

2x + 5 ≥ 11

Subtract 5 from both sides: 2x ≥ 6

Divide both sides by 2 (a positive number, so the inequality sign stays the same): x ≥ 3

Check:

• x = 3: 2(3) + 5 = 11, which is at least 11 ✓
• x = 4: 2(4) + 5 = 13 ✓
• x = 2: 2(2) + 5 = 9, which is less than 11 ✗

Answer: B. x ≥ 3

Tip: Watch the wording. "At least" means ≥, "at most" means ≤, "more than" means >, and "less than" means <. Also, remember to flip the inequality sign only when you multiply or divide by a negative number. Option A (x > 3) is a common trap because it leaves out x = 3, which is allowed.