Solving absolute value equation algebraically or graphically?

Question

Can this absolute value equation $$ | x - 2 | + | x - 1 | = 3 $$ be solved algebraically ? Or only graphically ? And, if yes, what is the solution ? Because, when I solved it algebraically, it gave me a different solution from the graphical one.

Answer 1

It can be. It can also be solved intuitively!

Use the fact: $|a-b|$ is the distance from $a$ to $b$. Then your equation translates as: "The distance from $x$ to $2$ plus the distance from $x$ to $1$ is $3$. Evidently, if $x$ is two units from $1$ and one unit from $2$, then it is a solution; looking at the number line I see $x = 3$ satisfies this condition; that makes it the first solution.

The other solution will be symmetric to that one; since $3$ is one unit to the right of the second point, then the other solution is one unit to the left of the first point, at $x = 0$.

The analytical solution is more work, but relies on the same idea! Imagine changing $x$ little by little, and think about how both of those distances change. When $x$ is very large, then its distance to $1$ and $2$ are very large. If $x$ decreases (moving left), then both distances will decrease for a while until $x$ reaches $2$. So the situation where $x > 2$ has one type of behavior. This corresponds to the sign of $x-2$!

Now when $x$ is less than $1$ and still moving left, both distances are increasing. This corresponds to the sign of $x-1$. So the analytical solution requires you to consider whether $x$ is

• to the left of both points,
• to the right of both points, or
• between the points.

In each of the cases, you know the sign of $x-1$ and $x-2$, and so you no longer need to use the absolute values. If $x-1$ is negative, then $|x-1|$ is equal to $-(x-1) = 1-x$, and so on.