Consider the sign of the number the absolute value is equal to.
Explanation: See solution. No Solution Example: |2x+1|=-1 One Solution Example: |2x+1|=0 Two Solutions Example: |2x+1|=1
Practice makes perfect
Let's consider each of the requested cases one at a time.
No Solution
The absolute value of any number number or expression is always non-negative, since it represents distance. Therefore, if we equate the absolute value of 2x+1 with a negative number, let's say - 1, we will have an equation with no solution.
|2x+1|=- 1
One Solution
We already know that an absolute value cannot be negative, so let's take a non-negative number p and set it equal to |2x+1|.
|2x+1|=p
When removing the absolute value from the equation, we need to consider two separate cases: a positive and a negative one.
2x+1=p or 2x+1=- p
In general, p and - p are always different numbers, except for the situation when p=0. In that case, the equations are equivalent.
l2x+1=0 2x+1=- 0 ⇒ 2x+1=0
The resulting equation 2x+1=0 has only one solution.
Two Solutions
Based on what we found earlier, an absolute value equation will have two solutions when the absolute value is equal to a positive number. Therefore, we can write an equation with two solutions by setting |2x+1| equal to any positive number, let's say 1.
|2x+1|=1
Keep in mind that all of these absolute value equations are just examples, and there are infinitely many other possible answers.
