Exercise 46 Page 33

Our friend is given the following absolute value equation. They say it has no solution because the constant on the right is negative.

∣ 3 x + 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.

∣ 3 x + 8 ∣ - 9 = - 5

AddEqn

$LHS + 9 = RHS + 9$

∣ 3 x + 8 ∣ = 4

$3 x + 8 \geq 0 : 3 x + 8 = 4 3 x + 8 < 0 : 3 x + 8 = - 4 (I) (II)$

3 x + 8 = 4 3 x + 8 = - 4 (I) (II)

$(I), (II):$ $LHS - 8 = RHS - 8$

3 x = - 4 3 x = - 12 (I) (II)

$(I), (II):$ $LHS / 3 = RHS / 3$

x_{1} = \frac{- 4}{3} x_{2} = - 4

MoveNegNumToFrac

$(I):$ Put minus sign in front of fraction

x_{1} = - \frac{4}{3} x_{2} = - 4

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.

∣ 3 x + 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.

∣ 3 x + 8 ∣ - 9 = - 5 \Leftrightarrow ∣ 3 x + 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.