The absolute value of a number is its distance from zero on the number line. So, an equation in the form of |x| = a represents all the numbers that are a units away from zero. An equation in the form of |x - b| = c represents all the numbers that are c units away from b on the number line.
To write an absolute value equation in the form x - b = c, we need to isolate the absolute value expression on one side of the equation and simplify the other side. Here's how we can do it for the given solution set:
Solution set: all numbers such that x <= 5
This means that the solution set includes all numbers less than or equal to 5. We can represent this graphically as a number line with a closed circle at 5 and an arrow pointing to the left.
To write this as an absolute value equation in the form x - b = c, we need to find the midpoint between 5 and 0, which is (5 + 0) / 2 = 2.5. Then, we can write:
|x - 2.5| = 2.5
This equation represents all the numbers that are 2.5 units away from 2.5 on the number line. These numbers include 0, 1, 2, 3, 4, and 5.
Alternatively, we can write:
|x - 5| = 0
This equation represents all the numbers that are 0 units away from 5 on the number line, which is just 5.
So, the two absolute value equations in the form x - b = c that have the solution set "all numbers such that x <= 5" are:
|x - 2.5| = 2.5
and
|x - 5| = 0