How to Solve Equations Involving Absolute Values?
By Robert O
An absolute value of any number is its distances from the origin to either side of the number line. That is, an absolute value can take negative or positive values. In writing, an absolute value of any number is by writing the number between two perpendicular bars. You should not interpret the bars as braces as they serve different purposes.
Mathematically, the definition of absolute values is
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How to solve equations involving absolute values?
There are steps that you need to follow to solve any equation that involves absolute values. We will list the steps and later use them to solve different examples:
Step 1: Isolation of absolute values on one side or both sides of the equation
Step 2: Check the sign of a number on one side if you don’t have absolute values on both sides. If the value is negative, then conclude that the equation has no solution to the equation. State that and abort the operation.
Step 3: Drop the bars for absolute values and then write two equations from the given one by setting values that were inside the bars equal to the value on the other side for the first equation. Equate the expression inside the bars to the opposite of the first equation to get the second equation.
Step 4: Solve the equations normally to get the two values of the unknown value.
Example 1:
Solve
![\[|x|=3\]](data:image/svg+xml;nitro-empty-id=MjUwOjMwMTk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTAgMTkiIHdpZHRoPSI1MCIgaGVpZ2h0PSIxOSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Solution
The absolute value is already on one side of the equation. We, therefore, skip step one.
Since we have a positive value on the other side, there is a solution to this problem.
Dropping the bars will result in two equations:
![\[x=3\]](data:image/svg+xml;nitro-empty-id=MjUwOjM3NDc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDIgMTIiIHdpZHRoPSI0MiIgaGVpZ2h0PSIxMiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[x=-3\]](data:image/svg+xml;nitro-empty-id=MjUwOjQxNzE=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTYgMTIiIHdpZHRoPSI1NiIgaGVpZ2h0PSIxMiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Example 2:
Solve
![\[|2 x-1|+3=5\]](data:image/svg+xml;nitro-empty-id=MjUwOjQ3MTM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTE5IDE5IiB3aWR0aD0iMTE5IiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
We have to first isolate the absolute value by taking +2 to the right side of the equation.
![\[|2 x-1|+2=5\]](data:image/svg+xml;nitro-empty-id=MjUwOjUzMTI=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTE5IDE5IiB3aWR0aD0iMTE5IiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[|2 x-1|=5-2\]](data:image/svg+xml;nitro-empty-id=MjUwOjU3NDQ=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTE5IDE5IiB3aWR0aD0iMTE5IiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[|2 x-1|=3\]](data:image/svg+xml;nitro-empty-id=MjUwOjYxNzM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgOTAgMTkiIHdpZHRoPSI5MCIgaGVpZ2h0PSIxOSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
The next step requires that we rewrite the resulting equation in two different ways and dropping the bars for absolute value. The two possible equations are:
![\[2 x-1=3 \ldots \ldots \ldots \ldots i\]](data:image/svg+xml;nitro-empty-id=MjUwOjY3OTU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTg2IDEyIiB3aWR0aD0iMTg2IiBoZWlnaHQ9IjEyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[2 x-1=-3 \ldots \ldots \ldots \ldots ii\]](data:image/svg+xml;nitro-empty-id=MjUwOjcyNTU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjA1IDEyIiB3aWR0aD0iMjA1IiBoZWlnaHQ9IjEyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solving equation (i) results into x=2 and equation (ii) results into x=-1. So, our solution for x gives 2 and -1, both of which satisfy the equation.
Example 3:
Solve
![\[|x-7|=|2 x-2|\]](data:image/svg+xml;nitro-empty-id=MjUwOjc5MjI=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTMwIDE5IiB3aWR0aD0iMTMwIiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
This equation has absolute values on both sides of the equation. We don’t need step one here and will jump to step two to see if there is a solution to the equation. The answer is yes, the equation has a solution.
Moving to step 3, we drop the bars on both sides and writing the equation in two different ways.
![\[x-7=2 x-2 \ldots \ldots \ldots \ldots \ldots i\]](data:image/svg+xml;nitro-empty-id=MjUwOjg3ODM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjQxIDEzIiB3aWR0aD0iMjQxIiBoZWlnaHQ9IjEzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[x-7=-(2 x-2) \ldots \ldots \ldots \ldots \ldots i i\]](data:image/svg+xml;nitro-empty-id=MjUwOjkyNTU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjc1IDE5IiB3aWR0aD0iMjc1IiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solving equation (i):
![\[x-7=2 x-2\]](data:image/svg+xml;nitro-empty-id=MjUwOjk3MTM=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTEzIDEzIiB3aWR0aD0iMTEzIiBoZWlnaHQ9IjEzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[x-2 x=-2+7\]](data:image/svg+xml;nitro-empty-id=MjUwOjEwMTQ0-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTI4IDE1IiB3aWR0aD0iMTI4IiBoZWlnaHQ9IjE1IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[-x=+5\]](data:image/svg+xml;nitro-empty-id=MjUwOjEwNTY5-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNjggMTUiIHdpZHRoPSI2OCIgaGVpZ2h0PSIxNSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[x=-5\]](data:image/svg+xml;nitro-empty-id=MjUwOjEwOTkz-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTUgMTMiIHdpZHRoPSI1NSIgaGVpZ2h0PSIxMyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Solving equation (ii):
![\[x-7=-(2 x-2)\]](data:image/svg+xml;nitro-empty-id=MjUwOjExNDU1-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTQxIDE5IiB3aWR0aD0iMTQxIiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[x-7=-2 x+2\]](data:image/svg+xml;nitro-empty-id=MjUwOjExODg2-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTI3IDE1IiB3aWR0aD0iMTI3IiBoZWlnaHQ9IjE1IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[x+2 x=7+2\]](data:image/svg+xml;nitro-empty-id=MjUwOjEyMzE2-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTEzIDE1IiB3aWR0aD0iMTEzIiBoZWlnaHQ9IjE1IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[3 x=9\]](data:image/svg+xml;nitro-empty-id=MjUwOjEyNzQx-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTIgMTIiIHdpZHRoPSI1MiIgaGVpZ2h0PSIxMiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[x=3\]](data:image/svg+xml;nitro-empty-id=MjUwOjEzMTY0-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDIgMTIiIHdpZHRoPSI0MiIgaGVpZ2h0PSIxMiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Both solutions, x=-5 and x=3, are the values that we have been looking for, and they satisfy the equation.
Example 4:
Solve
![\[|x-3|+6=|3-x|+6\]](data:image/svg+xml;nitro-empty-id=MjUwOjEzNzkw-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTg0IDE5IiB3aWR0aD0iMTg0IiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Solution
This equation has absolute values on both sides that need isolation. We can do that by subtracting 6 from both sides of the equation to get:
![\[|x-3|=|3-x|\]](data:image/svg+xml;nitro-empty-id=MjUwOjE0NDM4-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTIwIDE5IiB3aWR0aD0iMTIwIiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Next, we write two equations from the original one and dropping the bars.
![\[x-3=3-x \ldots \ldots \ldots \ldots i\]](data:image/svg+xml;nitro-empty-id=MjUwOjE0OTc2-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjA4IDEyIiB3aWR0aD0iMjA4IiBoZWlnaHQ9IjEyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[x-3=-(3-x) \ldots \ldots \ldots \ldots \ldots ii\]](data:image/svg+xml;nitro-empty-id=MjUwOjE1NDQ1-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY2IDE5IiB3aWR0aD0iMjY2IiBoZWlnaHQ9IjE5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
We leave that for you to solve and confirm that the solutions that we need to satisfy the above equation are all real numbers. The first equation will give x=3, and the second question results in -3=-3, from which we deduce that all real numbers are the solution to the equation.
Note: If after solving any of the equation and the result is a false statement like 2=3, you conclude that the equation has no solution.
Remarks
Working with equations with absolute values is pretty straightforward. There is no complex algebra in it to take your time, but the knowledge is useful in solving real-life problems.
About the Author
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical engineering and electronic engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.