Our friend is given the following absolute value equation. They say it has no solution because the constant on the right is negative.
|3x+8|-9=- 5
To see if they are right, let's try to solve the equation. First, we need to isolate the absolute value expression.
We see that our friend is incorrect since there are solutions to the equation. The reason is they have misunderstood a key piece of the idea. The absolute value can be thought of as distance. Therefore, it must always be positive. Consider the following absolute value equation.
| x+2|= - 3
The equation has no solution because it shows the absolute value of an expression x+2 equal to - 3, which is a negative number. However, notice that on the left-hand side, the absolute value is isolated. Let's now recall the equation given in the exercise.
|3x+8|-9=- 5
Although the left-hand side is set equal to a negative number, the absolute value is not isolated. That is why we cannot conclude that there is no solution. As a first step, when solving, we isolated the absolute value.
|3x+8|-9=- 5 ⇔ |3x+8|= 4
Now we can see that absolute value of an expression is equal to 4, which is a positive number. This means that the equation has solutions, which we found earlier.
