Exercise 57 Page 34

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 $\text{-}1,$ we will have an equation with no solution.

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 $|2x+1|.$ When removing the absolute value from the equation, we need to consider two separate cases: a positive and a negative one. In general, $p$ and $\text{-}p$ are always different numbers, except for the situation when $p=0.$ In that case, the equations are equivalent. 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 $1.$ Keep in mind that all of these absolute value equations are just examples, and there are infinitely many other possible answers.