The solutions for |x| ≤ a are the numbers whose distance from 0 is less than or equal to a. The solutions for |x| ≥ a are the numbers whose distance from 0 is greater than or equal to a.
Solutions for |x|+2≤5: x ≥ - 3 and x ≤ 3 Solutions for |x|+2≥ 5: x ≥ 3 or x ≤ - 3
Practice makes perfect
We want to describe the solutions to the inequalities |x|+2 ≤ 5 and |x|+2 ≥ 5 without using absolute value. Let's do it one at a time.
Solutions of |x|+2 ≤ 5
We will start by isolating the absolute value expression.
Recall now that the absolute value of a number x is its distance from 0. Therefore, the above means that the distance from 0 to x is less than or equal to 3. The numbers whose distance from 0 is less than or equal to 3 are those numbers that are greater than or equal to - 3, andless than or equal to 3.
Using the above graph, we can write two inequalities to represent the situation.
x ≥ - 3 AND x ≤ 3
Note that the word and is used because x must be both greater than or equal to - 3 and less than or equal to 3.
Solutions of |x|+2 ≥ 5
Let's now solve the second inequality. Again, we will start by isolating the absolute value expression.
One more time, recall that the absolute value of a number x is its distance from 0. Therefore, the above means that the distance from 0 to x is greater than or equal to 3. The numbers whose distance from 0 is greater than or equal to 3 are those numbers that are greater than or equal to 3, orless than or equal to - 3.
Using the above graph, we can write two inequalities to represent the situation.
x ≥ 3 OR x ≤ - 3
Note that the word or is used because x must be either greater than or equal to 3 or less than or equal to - 3.
