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.

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 if x is non-negative; and
• |x| = -x if x is negative.

For example:

• |4| = 4;
• |0| = 0; and
• |-4| = 4.

In other words, the absolute value of a non-negative number is exactly this number, while for a negative number you have to throw away the minus sign.

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!

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:

1. Enter the coefficients a, b, c, d, e of your absolute value equality. Remember that neither a nor b can be equal to zero - we don't want x to disappear from the equation!
2. 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.

If you want to solve by hand an absolute value equation of the form a * |bx + c| + d = e, follow these steps:

1. Simplify your equation: transfer d to the right-hand side and divide both sides by a. You'll get |bx + c| = (e - d)/a.
2. Look at the sign on the right-hand 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 to -c/b; and
   • If (e - d)/a > 0, then your equation has two solutions. Let's find them.
3. Omitting the absolute value, we have bx + c = (e - d)/a or bx + c = -(e - d)/a. It now suffices to calculate x for each equation.
4. Finally, x = (e - d)/(a * b) - c/b or x = (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.

Graphing absolute value equations of the form a * |bx + c| + d = e is simple, you only need to remember a few basic rules:

1. Start by plotting bx + c. It is a straight line that contains the points (0,c) and (-c/b,0).
2. Reflect the negative part of this line (i.e., where y<0) through the horizontal axis to get |bx + c|.
3. Multiplying by a is equivalent to changing the slope. Additionally, if a<0, you have to reflect your plot one more time through the horizontal axis.
4. Adding d boils down to moving the plot upwards or downwards, depending on whether d is positive or negative. In any case, you now have the plot of a * |bx + c| + d.
5. Draw the line y = e. The points where the two graphs intersect (if there are any) are the solutions of your equations.

Example 1. Solve |2x + 5| = x + 4

1. We resolve the absolute value on the left-hand side. Clearly 2x + 5 ≥ 0 is equivalent to x ≥ -2.5 Consequently, we obtain
   -2x - 5 = x + 4 if x < -2.5
   and
   2x + 5 = x + 4 if x ≥ -2.5

2. Simplifying, we get
   -3x = 9 if x < -2.5
   and
   x = 1 if x ≥ -2.5

3. As a result, we have
   x = -3 or x = 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 right-hand side:
|x| = -x + 1 + 1 if x ≥ 1
and
|x| = x - 1 + 1 if x < 1

2. Simplify:
   |x| = -x + 2 if x ≥ 1
   and
   |x| = x if x < 1

3. We will solve the equation |x| = -x + 2 if x ≥ 1. Resolve the absolute value on the left-hand side:

   • x = -x + 2 if x ≥ 1 and x ≥ 0
   • -x = -x + 2 if x ≥ 1 and x < 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.

4. Solution of the first equation: 2x = 2 if x ≥ 1. So x = 1 is indeed a solution.

5. Next, we will solve the equation |x| = x if x < 1. Resolve the absolute value on the left-hand side:

   • x = x if x < 1 and x ≥ 0
   • -x = x if x < 1 and x < 0

6. Simplify:

   • 0 = 0 if 0 ≤ x < 1
   • 2x = 0 if x < 0

7. Solve:

   • All x ∈ [0 , 1) satisfy the equation in question
   • x = 0 does not satisfy the condition x < 0, so this is not a solution.

   So all x in the interval [0,1) are solutions to the equation |x| = x if x < 1.

8. Final solution: We found that x = 1 and x ∈ [0 , 1) are solutions. Taking all of this together, we see that every x ∈ [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!

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.

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 non-negative.

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 non-zero number will always have its absolute value equal to a positive number!