Question

use set-builder notation to describe all real numbers satisfying the given conditions.
a number increased by 7 is at least two times the number.

Explanation:

Step1: Define the variable

Let the number be \( x \).

Step2: Translate the condition into an inequality

"A number increased by 7" is \( x + 7 \), "at least two times the number" is \( 2x \). So the inequality is \( x + 7 \geq 2x \).

Step3: Write the set - builder notation

The set - builder notation for all real numbers \( x \) satisfying \( x + 7 \geq 2x \) is \( \{x\in\mathbb{R}\mid x + 7\geq2x\} \). We can also simplify the inequality:
Subtract \( x \) from both sides of \( x + 7\geq2x \), we get \( 7\geq x \) or \( x\leq7 \). So the set - builder notation can also be written as \( \{x\in\mathbb{R}\mid x\leq7\} \)

Answer:

\(\{x\in\mathbb{R}\mid x + 7\geq2x\}\) (or \(\{x\in\mathbb{R}\mid x\leq7\}\))