Absolute Value Equation Calculator
Whenever you face absolute value equations, Omni's absolute value equation calculator is here to lend you a hand. With its help, you'll easily deal with all kinds of absolute value equalities, in particular equations where the absolute value is equal to 0. If you want to learn more about this topic, scroll down and learn:
 How to solve absolute value equations by hand; and
 How to find an absolute value equation from a graph.
Once you're done here, take one step further in your mathematical journey and check out our absolute value inequalities calculator.
What are absolute value equalities?
Let's briefly recall what the absolute value is, shall we? The absolute value of a real number x
is the distance between this number and zero. We denote it by x
. Algebraically:
x = x
ifx
is nonnegative; andx = x
ifx
is negative.
For example:
4 = 4
;0 = 0
; and4 = 4
.
In other words, the absolute value of a nonnegative number is exactly this number, while for a negative number you have to throw away the minus sign.
💡 Did you know that absolute value is very popular in statistics? You can find it, e.g., in the following notions:
We can now move on and discuss what the absolute value equations are.
Absolute value equalities
In general, any equation involving the absolute value of any expression is an absolute value equality. It can involve, e.g., polynomial expressions, roots, exponents, logarithms, etc.
In school, you're most likely to encounter an absolute value equation of the linear expression bx+c
, that is, an equation of the form
a * bx + c + d = e
,
where a, b, c, d, e
are real coefficients. And here's where our absolute value equation calculator enters the stage!
How to use this absolute value equation calculator?
Omni's absolute value equation calculator can help you solve absolute value equalities of the form a * bx + c + d = e
. Here's a brief instruction on how to solve them most efficiently:
 Enter the coefficients
a, b, c, d, e
of your absolute value equality. Remember that neithera
norb
can be equal to zero  we don't wantx
to disappear from the equation!  Your equation will appear at the bottom of the calculator  verify that
everything is all right.
3. Below the equation, you'll see the solution to your absolute value equation as well! By default, the calculator uses 4 decimal places to display the solution. Click the Advanced mode
to adjust the precision.
4. Turn the Show steps?
option to Yes
to see some intermediate computations performed by our tool.
5. Turn the Show graph?
option to Yes
to see how to graph your absolute value equation.
How to solve absolute value equations by hand?
If you want to solve by hand an absolute value equation of the form a * bx + c + d = e
, follow these steps:
 Simplify your equation: transfer
d
to the righthand side and divide both sides bya
. You'll getbx + c = (e  d)/a
.  Look at the sign on the righthand side:
 If
(e  d)/a < 0
, then you've got an absolute value equation with no solutions;  If
(e  d)/a = 0
, then you've got an absolute value equation with one solution, which is equal toc/b
; and  If
(e  d)/a > 0
, then your equation has two solutions. Let's find them.
 If
 Omitting the absolute value, we have
bx + c = (e  d)/a
orbx + c = (e  d)/a
. It now suffices to calculatex
for each equation.  Finally,
x = (e  d)/(a * b)  c/b
orx = (d  e)/(a * b)  c/b
.
As you can see, it's not at all hard to solve absolute value equations by hand. Practice a bit with our absolute value equation calculator, and you'll quickly become an expert! As the next step, you may want to learn how to find an absolute value equation from a graph.
How to graph absolute value equations?
Graphing absolute value equations of the form a * bx + c + d = e
is simple, you only need to remember a few basic rules:
 Start by plotting
bx + c
. It is a straight line that contains the points(0,c)
and(c/b,0)
.  Reflect the negative part of this line (i.e., where
y<0
) through the horizontal axis to getbx + c
.  Multiplying by
a
is equivalent to changing the slope. Additionally, ifa<0
, you have to reflect your plot one more time through the horizontal axis.  Adding
d
boils down to moving the plot upwards or downwards, depending on whetherd
is positive or negative. In any case, you now have the plot ofa * bx + c + d
.  Draw the line
y = e
. The points where the two graphs intersect (if there are any) are the solutions of your equations.
Once you're done with learning how to graph absolute value equations, take it a step further and learn about graphing polynomials! But, before that, make sure to take a look at a few examples of absolute value equations.
Examples of absolute value equations
Example 1. Solve 2x + 5 = x + 4

We resolve the absolute value on the lefthand side. Clearly
2x + 5 ≥ 0
is equivalent tox ≥ 2.5
Consequently, we obtain
2x  5 = x + 4
ifx < 2.5
and
2x + 5 = x + 4
ifx ≥ 2.5

Simplifying, we get
3x = 9
ifx < 2.5
and
x = 1
ifx ≥ 2.5

As a result, we have
x = 3
orx = 1
You can use our absolute value solver to graph this solution.
Example 2. Solve x = x  1 + 1
1.We resolve the absolute value on the righthand side:
x = x + 1 + 1
if x ≥ 1
and
x = x  1 + 1
if x < 1

Simplify:
x = x + 2
ifx ≥ 1
and
x = x
ifx < 1

We will solve the equation
x = x + 2
ifx ≥ 1
. Resolve the absolute value on the lefthand side:x = x + 2
ifx ≥ 1
andx ≥ 0
x = x + 2
ifx ≥ 1
andx < 0
Note, that in the second equation we have a contradiction in the conditions. So we ignore this equation and only solve the first one.

Solution of the first equation:
2x = 2
ifx ≥ 1
. Sox = 1
is indeed a solution. 
Next, we will solve the equation
x = x
ifx < 1
. Resolve the absolute value on the lefthand side:x = x
ifx < 1
andx ≥ 0
x = x
ifx < 1
andx < 0

Simplify:
0 = 0
if0 ≤ x < 1
2x = 0
ifx < 0

Solve:
 All
x ∈ [0 , 1)
satisfy the equation in question x = 0
does not satisfy the conditionx < 0
, so this is not a solution.
So all
x
in the interval[0,1)
are solutions to the equationx = x
ifx < 1
.  All

Final solution: We found that
x = 1
andx ∈ [0 , 1)
are solutions. Taking all of this together, we see that everyx ∈ [0 , 1]
is a solution to our problem.
With the help of our absolute value solver, you can train yourself to solve this kind of problem, and become a master of absolute value equations!
FAQ
How many solutions can an absolute value equation have?
The number of solutions of an absolute value equation of the form a * bx + c + d = e
depends on the sign of (e  d)/a
. Namely, we have:
 Two solutions if
(e d)/a > 0
;  One solution if
e = d
; and  No solution if
(e d)/a < 0
.
Can an absolute value be equal to a negative number?
No, absolute values cannot be equal to negative numbers. This follows from the very definition of the absolute value of a number, which is the distance between zero and this number, and distances, as we all know, are always nonnegative.
Can an absolute value be equal to 0?
Yes, but in a very special case. Namely, the only situation where the absolute value of a number is zero is when this number is zero: x = 0
if and only if x = 0
. A nonzero number will always have its absolute value equal to a positive number!