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

2 x + 1

with a negative number, let's say

- 1,

we will have an equation with no solution.

∣ 2 x + 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

∣ 2 x + 1 ∣ .

∣ 2 x + 1 ∣ = p

When removing the absolute value from the equation, we need to consider two separate cases: a positive and a negative one.

2 x + 1 = p or 2 x + 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.

2 x + 1 = 0 2 x + 1 = - 0 \Rightarrow 2 x + 1 = 0

The resulting equation

2 x + 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

∣ 2 x + 1 ∣

equal to any positive number, let's say

1 .

∣ 2 x + 1 ∣ = 1

Keep in mind that all of these absolute value equations are just examples, and there are infinitely many other possible answers.