Question

3 use distance on a number line to explain why the solutions to |3 - x| ≥ 2 include all numbers less than or equal to 1 as well as all numbers greater than or equal to 5.

4 a. explain what the solutions to the inequality |x - 4.26| ≥ 0 mean based on distance on a number line. what are the solutions?

b. explain what the solutions to the inequality |x - 2| < 0 mean based on distance on a number line. what are the solutions?
the solution of 2|x + 3| < 10 is -8 < x < 2

Explanation:

4a

The absolute value \(|x - 4.26|\) represents the distance between \(x\) and \(4.26\) on the number line. Since distance is always non - negative (i.e., \(|a|\geq0\) for any real number \(a\)), the inequality \(|x - 4.26|\geq0\) means that all real numbers \(x\) satisfy this inequality. Because for any real number \(x\), the distance between \(x\) and \(4.26\) is either \(0\) (when \(x = 4.26\)) or a positive number.

The absolute value \(|x - 2|\) represents the distance between \(x\) and \(2\) on the number line. By the definition of absolute value, \(|a|\geq0\) for any real number \(a\). So, the inequality \(|x - 2|\lt0\) has no solution. Because there is no real number \(x\) such that the distance between \(x\) and \(2\) (which is \(|x - 2|\)) is less than \(0\).

Answer:

All real numbers are solutions.

4b