Question

How can an absolute value equation have one solution?

Answer 1

Hint: We are given an absolute expression. We have to determine the method to identify whether it has one solution. First, determine the term at the right hand side of the equation. If the right hand side contains zero, then it has one solution. If it contains any other positive value, then the number of solutions is two. Otherwise there is no solution.

Complete step by step solution:
The absolute value equations are written as variables enclosed within an absolute value operator equated by some constant at the right hand side.

For example, $\left| {x + 5} \right| = 4$

When the absolute value equation is compared with any positive number, then the number of solutions are two.

For example, in the expression $\left| {x + 5} \right| = 4$. For this equation to be true

Either

$ \Rightarrow x + 5 = 4$
Or
$ \Rightarrow x + 5 = - 4$

On solving first equation, $x + 5 = 4$ we get:

$ \Rightarrow x + 5 - 5 = 4 - 5$

$ \Rightarrow x = - 1$

On solving second equation, $x + 5 = - 4$ we get:

$ \Rightarrow x + 5 - 5 = - 4 - 5$

$ \Rightarrow x = - 9$

Therefore, there are two solutions of the absolute value equation.

When the right hand side of the equation is zero, then the absolute value equation has only one solution.

For example, consider an equation $\left| {x + 5} \right| = 0$

For this equation to be true

$ \Rightarrow x + 5 = 0$

On solving the equation, we get:

$ \Rightarrow x + 5 - 5 = 0 - 5$

$ \Rightarrow x = - 5$

Therefore, the absolute value equation has only one solution.

Hence the absolute value equation has only one solution when it is compared to 0.

Note: The students must note that the absolute values represents the distance from zero on the number line, such as in the above equation $\left| {x + 5} \right| = 4$ it represents that the distance of $x + 5$ from zero on the number line is equal to 4 units. Therefore, the absolute values are always positive values.