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Absolute Value Equations with No Solution

Earlier, we looked at absolute value equations with sums and with products. We also looked at other approaches. Now, let's look at another option: no solutions.

There's one MAJOR red flag of an equation with an absolute value that has no solution.

Recall that an absolute value expression can never be less than zero. That is, a fully reduced absolute value expression must be greater than or equal to zero.

If you've simplified an absolute value equation and the value on the other side of the equals sign is a negative number, the equation has no solution.

Let's look at an example:

$$2|3x + 1| + 6 = 3$$

First, we need to simplify this equation as much as possible, isolating the absolute value expression, like we isolate the $x$ value when solving for $x$.

$$\begin{align} 2|3x + 1| + 6 &= 3 \\ 2|3x + 1| &= -3 \\ |3x + 1| &= -\frac{3}{2} \\ \end{align}$$

Now that we've isolated the absolute value expression, we can see that it's been set equal to a negative number. This is just not possible. By definition, an absolute value is greater than or equal to $0$.

Anytime an equation makes a false statement, it has no solution.

Sometimes absolute value equations don't have a solution. This usually happens when the absolute value equals a negative number. For example: $|x|=-3$. No matter what value you put in for $x$ you will never get $-3$ as an answer.

Above is the graph of $y=|x|$. Notice that it never goes below $0$. So the only time there is no solution is when $y<0$.

A note!

Keep in mind that you must solve the equation as much as you can first. For example:

$$|x-3|-6=-5$$

From first glance you might assume it has no solution since there's a $-5$ at the end. HOWEVER, try adding $6$ to both sides:

$$|x-3|=1$$

This has a the absolute value equal to a positive number so it has a solution.

Let's change that equation up a bit. Try:

$$|x-3|-5=-6$$

If we add $5$ to each side we get:

$$|x-3|=-1$$

This does not have a solution because it's an absolute value equal to a negative number.

(Video) Solving an Absolute Value Equation with no Solution - Free Math Videos

by Brian McLogan

This video by Brian McLogan looks at absolute value equations with no solution.

$\large{\text{Summary}}$

Absolute values must be positive or $0$. This means absolute values set equal to a negative number have no solution. Let's look at an example

$$|-5-4x|+9=-40$$

First, we have to isolate the absolute value sign. We'll subtract $9$ from both sides.

$$|-5-4x|=-49$$

If the number on the right hand side of the equation was positive, say $49,$ we would set the absolute value equal $\pm 49$ and solve for $x$ as we've previously done. However, the absolute value is equal to a negative number, so there is no solution.