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• High School Math Solutions – Absolute Value Equation Calculator Solving absolute value equations is somewhat tricky; it requires understanding of the absolute value property.... Read More

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To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations.

Solve | x | + 2 = 5.

Isolate the absolute value.

Set the contents of the absolute value portion equal to +3 and –3.

Solve 3| x – 1| – 1 = 11.

Set the contents of the absolute value portion equal to +4 and –4.

Solving for x,

Solving inequalities containing absolute value and graphing

To solve an inequality containing absolute value , begin with the same steps as for solving equations with absolute value. When creating the comparisons to both the + and – of the other side of the inequality, reverse the direction of the inequality when comparing with the negative.

Solve and graph the answer: | x – 1| > 2.

Notice that the absolute value expression is already isolated.

| x – 1| > 2

Compare the contents of the absolute value portion to both 2 and –2. Be sure to reverse the direction of the inequality when comparing it with –2.

Solve for x .

Graph the answer (see Figure 1).

Figure 1. The graphic solution to | x – 1| > 2.

Solve and graph the answer: 3| x | – 2 ≤ 1.

Compare the contents of the absolute value portion to both 1 and –1. Be sure to reverse the direction of the inequality when comparing it with –1.

Graph the answer (see Figure 2).

Figure 2. Graphing the solution to 3| x | – 2 ≤ 1.

Solve and graph the answer: 2|1 – x | + 1 ≥ 3.

(Remember to switch the direction of the inequality when dividing by a negative)

Graph the answer (see Figure 3).

Figure 3. Graphing the solution 2|1 – x | + 1 ≥ 3.

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Solving Simpler Absolute-Value Equations

Simpler Harder Special Case

When we take the absolute value of a number, we always end up with a positive number (or zero). Whether the input was positive or negative (or zero), the output is always positive (or zero). For instance, | 3 | = 3 , and | −3 | = 3 also.

This property — that both the positive and the negative become positive — makes solving absolute-value equations a little tricky. But once you learn the "trick", they're not so bad. Let's start with something simple:

Content Continues Below

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Solving Absolute Value Equations

Solve | x | = 3

I've pretty much already solved this, in my discussion above:

| −3 | = 3

So then x must be equal to 3 or equal to −3 .

But how am I supposed to solve this if I don't already know the answer? I will use the positive / negative property of the absolute value to split the equation into two cases, and I will use the fact that the "minus" sign in the negative case indicates "the opposite sign", not "a negative number".

For example, if I have x = −6 , then " − x " indicates "the opposite of x " or, in this case, −(−6) = +6 , a positive number. The "minus" sign in " − x " just indicates that I am changing the sign on x . It does not indicate a negative number. This distinction is crucial!

Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might originally have been positive and might originally have been negative, I must acknowledge this fact when I remove the absolute-value bars. I do this by splitting the equation into two cases. For this exercise, these cases are as follows:

a. If the value of x was non-negative (that is, if it was positive or zero) to start with, then I can bring that value out of the absolute-value bars without changing its sign, giving me the equation x = 3 .

b. If the value of x was negative to start with, then I can bring that value out of the absolute-value bars by changing the sign on x , giving me the equation − x = 3 , which solves as x = −3 .

Then my solution is

x = ±3

We can, by the way, verify the above solution graphically. When we attempt to solve the absolute-value equation | x  | = 3 , we are, in effect, setting two line equations equal to each other and finding where they cross. For instance:

In the above, I've plotted the graph of y 1  = |  x  | (being the blue line that looks like a "V") and y 2  = 3 (being the green horizontal line). These two graphs cross at x  = −3 and at x  = +3 (being the two red dots).

If you're wanting to check your answers on a test (before you hand it in), it can be helpful to plug each side of the original absolute-value equation into your calculator as their own functions; then ask the calculator for the intersection points.

Of course, any solution can also be verified by plugging it back into the original exercise, and confirming that the left-hand side (LHS) of the equation simplifies to the same value as does the right-hand side (RHS) of the equation. For the equation above, here's my check:

x = −3

LHS: | x | = | −3 |

LHS: | x | = | +3 |

If you're ever in doubt about your solution to an equation, try graphing or else try plugging your solution back into the original question. Checking your work is always okay!

The step in the above, where the absolute-value equation was restated in two forms, one with a "plus" and one with a "minus", gives us a handy way to simplify things: When we have isolated the absolute value and go to take off the bars, we can split the equation into two cases; we will signify these cases by placing a "minus" on the opposite side of the equation (for one case) and a "plus" on the opposite side (for the other). Here's how this works:

Solve | x + 2 | = 7 , and check your solution(s).

The absolute value is isolated on the left-hand side of the equation, so it's already set up for me to split the equation into two cases. To clear the absolute-value bars, I must split the equation into its two possible two cases, one each for if the contents of the absolute-value bars (that is, if the "argument" of the absolute value) is negative and if it's non-negative (that is, if it's positive or zero). To do this, I create two new equations, where the only difference between then is the sign on the right-hand side. First, I'll do the "minus" case:

x + 2 = −7

x = −9

Now I'll do the non-negative case, where I can just drop the bars and solve:

Now I need to check my solutions. I'll do this by plugging them back into the original equation, since the grader can't see me checking plots on my graphing calculator.

x = −9:

LHS: |(−9) + 2|

= |−7| = 7 = RHS

LHS: |(5) + 2|

= |7| = 7 = RHS

Both solutions check, so my answer is:

x = −9, 5

Solve | 2 x − 3 | − 4 = 3

First, I'll isolate the absolute-value part of the equation; that is, I'll get the absolute-value expression by itself on one side of the "equals" sign, with everything else on the other side:

| 2 x − 3 | − 4 = 3

| 2 x − 3 | = 7

Now I'll clear the absolute-value bars by splitting the equation into its two cases, one for each sign on the argument. First I'll do the negative case:

2 x − 3 = −7

2 x = −4

x = −2

And then I'll do the non-negative case:

2 x − 3 = 7

The exercise doesn't tell me to check, so I won't. (But, if I'd wanted to, I could have plugged "abs(2X−3)−4" and "3" into my calculator (as Y1 and Y2, respectively), and seen that the intersection points were at my x -values.) My answer is:

x = −2, 5

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How to Solve Absolute Value Equations

Last Updated: January 31, 2023 References

This article was co-authored by wikiHow Staff . Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 23,117 times.

Setting up the Problem

• For example, |9| = 9; |-9| = -(-9) = 9.

Expert Q&A

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• Remember that absolute value bars are distinct from parentheses and function differently. Thanks Helpful 0 Not Helpful 0
• Once you've solved for any variables, remember to simplify absolute values accordingly. Thanks Helpful 0 Not Helpful 0

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• ↑ http://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueEqns.aspx
• ↑ https://www.mathsisfun.com/numbers/absolute-value.html
• ↑ http://www.varsitytutors.com/high_school_math-help/solving-absolute-value-equations
• ↑ http://www.purplemath.com/modules/solveabs.htm

To solve absolute value equations, first isolate the absolute value terms by moving anything outside of the vertical bars to the other side of the equation. Next, solve for the positive value of the equation by isolating the variable. Since the absolute variable can represent 2 numbers, then solve for the negative value by putting a negative sign outside the vertical bars. Then, move the negative by dividing both sides by -1 and solve for the variable. If you want to learn how to check your answers for an absolute value equation, keep reading the article! Did this summary help you? Yes No

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Optimization with absolute values

Authors: Matthew Chan (mdc297), Yilian Yin (yy896), Brian Amado (ba392), Peter Williams (pmw99), Dewei Xiao (dx58) (SysEn 5800 Fall 2020)

• 1 Introduction
• 2.1 Defining Absolute Values

• 2.3.1 Sign Constraints are Satisfied
• 2.3.2 Sign Constraints are Not Satisfied
• 2.4 Absolute Values in Nonlinear Optimization Problems
• 3 Numerical Example
• 4.1 Application in Financial: Portfolio Selection
• 4.2 Data Transfer Rate
• 5 Conclusion
• 6 References

Defining Absolute Values

An absolute value of a real number can be described as its distance away from zero, or the non-negative magnitude of the number. [1] Thus,

Absolute values can exist in linear optimization problems in two primary instances: in constraints and in the objective function. [2]

Absolute Values in Constraints

Within constraints, absolute value relations can be transformed into one of the following forms:

Form when | X | ≥ C {\displaystyle \displaystyle |X|\geq C}

Visually, the solution space for the last form is the complement of the second solution above, resulting in the following representation:

In expression form, the solutions can be written as:

As seen visually, the feasible region has a gap and thus non-convex. The expressions also make it impossible for both to simultaneously hold true. This means that it is not possible to transform constraints in this form to linear equations. [3]

An approach to reach a solution for this particular case exists in the form of Mixed-Integer Linear Programming, where only one of the equations above is “active”.

The inequality can be reformulated into the following:

Absolute Values in Objective Functions

In objective functions, to leverage transformations of absolute functions, all constraints must be linear.

Similar to the case of absolute values in constraints, there are different approaches to the reformation of the objective function, depending on the satisfaction of sign constraints. The satisfaction of sign constraints is when the coefficient signs of the absolute terms must all be either:

• Positive for a minimization problem
• Negative for a maximization problem

Sign Constraints are Not Satisfied

In order to transform problems where the coefficient signs of the absolute terms do not fulfill the conditions above, a similar conclusion is reached to that of the last case for absolute values in constraints – the use of integer variables is needed to reach an LP format.

The following constraints need to be added to the problem:

Numerical Example

Example when All Sign Constraints are Satisfied

The absolute value quantities will be replaced with single variables:

We must introduce additional constraints to ensure we do not lose any information by doing this substitution:

The problem has now been reformulated as a linear programming problem that can be solved normally:

Example when Sign Constraints are not Satisfied

The problem has now been reformulated as a linear programming problem that can be solved normally: [4]

Applications

The primary application of absolute-value functionals in linear programming has been for absolute-value or L(i)-metric regression analysis. Such application is always a minimization problem with all C(j) equal to 1 so that the required conditions for valid use of the simplex method are met.

By reformulating the original problem into a Mixed-Integer Linear Program (MILP), we can utilize known programs to solve for the optimal solution(s).

Application in Financial: Portfolio Selection

Under this topic, the same tricks played in the Numerical Example section to perform Reduction to a Linear Programming Problem will be applied here again, to reform the problem into a MILP in order to solve the problem. An example is given as below.

So finally, after some simplifications methods and some tricks applied, the original problem is converted into a linear programming which is easier to be solved further.

Data Transfer Rate

Another application of optimization with absolute values is data transfer rate. Faster-than-nyquist, or FTNS, is a framework to transmit signals beyond the Nyquist rate. The refence to this section proposed a 24.7% faster symbol rate by utilizing Sum-of-Absolute-Values optimization. [6]

The presence of an absolute value within the objective function prevents the use of certain optimization methods. Solving these problems requires that the function be manipulated in order to continue with linear programming techniques like the simplex method. The applications of optimization with absolute values range from the financial sector to the digital world where data transfer rates can be improved as well as improving portfolio returns. The way these problems are formulated, must take absolute values into account in order to model the problem correctly. The absolute values inherently make these problems non-linear so determining the most optimal solutions is only achievable after reformulating them into linear programs.

• ↑ Mendelson, Elliott, Schaum's Outline of Beginning Calculus, McGraw-Hill Professional, 2008. https://books.google.com/books?id=A8hAm38zsCMC&pg=PA2#v=onepage&q&f=false
• ↑ "Absolute Values." lp_solve , http://lpsolve.sourceforge.net/ . Accessed 20 Nov. 2020.
• ↑ Optimization Methods in Management Science / Operations Research. Massachusetts Institute of Technology, Spring 2013, https://ocw.mit.edu/courses/sloan-school-of-management/15-053-optimization-methods-in-management-science-spring-2013/tutorials/MIT15_053S13_tut04.pdf . Accessed 20 Nov. 2020.
• ↑ Shanno, David F., and Roman L. Weil. “'Linear' Programming with Absolute-Value Functionals.” Operations Research, vol. 19, no. 1, 1971, pp. 120–124. Accessed 13 Dec. 2020. JSTOR, www.jstor.org/stable/168871.
• ↑ Vanderbei R.J. (2008) Financial Applications. In: Linear Programming. International Series in Operations Research & Management Science, vol 114. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74388-2_13 https://link.springer.com/chapter/10.1007/978-0-387-74388-2_13
• ↑ Sasahara, Hampei & Hayashi, Kazunori & Nagahara, Masaaki. (2016). Symbol Detection for Faster-Than-Nyquist Signaling by Sum-of-Absolute-Values Optimization. IEEE Signal Processing Letters. PP. 1-1. 10.1109/LSP.2016.2625839. https://www.researchgate.net/publication/309745511_Symbol_Detection_for_Faster-Than-Nyquist_Signaling_by_Sum-of-Absolute-Values_Optimization

IMAGES

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2. Solving Absolute Value Equations (solutions, examples, videos

3. how to solve a problem with absolute value

4. Objective

5. how to solve a problem with absolute value

6. How To Solve Absolute Value Equations

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1. 5301 Solving Absolute Value Equations 02

2. 1.4 Absolute Value Equations

3. ABSOLUTE VALUE EQUATIONS(SECTI0N 1.8)

4. 4. Solving Absolute Value Pt 2

5. absolute value equations 23 24

6. How to solve this nice algebra equation |x=?

1. Absolute Value Equation Calculator

Free absolute value equation calculator - solve absolute value equations with all the steps. Type in any equation to get the solution, steps and graph

2. Absolute value word problems (video)

Learn to solve word problems that ask us to apply our new absolute value knowledge. Created by Sal Khan. Questions Tips & Thanks Want to join the conversation? Sort by: Top Voted aubrey 10 years ago Is there a term for the little lines indicating absolute value? • 5 comments ( 39 votes) Upvote Downvote Flag

3. Solving Equations Containing Absolute Value

To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations. Solve | x | + 2 = 5. Isolate the absolute value.

4. Solving Simpler Absolute-Value Equations

When we attempt to solve the absolute-value equation | x | = 3, we are, in effect, setting two line equations equal to each other and finding where they cross. For instance: In the above, I've plotted the graph of y1 = | x | (being the blue line that looks like a "V") and y2 = 3 (being the green horizontal line).

5. Algebra

Inequalities Involving > and ≥ ≥. Once again let’s start off with a simple number example. |p| ≥ 4 | p | ≥ 4. This says that whatever p p is it must be at least a distance of 4 from the origin and so p p must be in one of the following two ranges, p ≤ −4 or p ≥ 4 p ≤ − 4 or p ≥ 4.

6. How to Solve Absolute Value Equations: 10 Steps (with Pictures)

To check your positive equation, plug the value for derived from the positive equation back into the original absolute value equation. If both sides of the equation are equal, the solution is true. For example, if the solution to the positive equation was. x = 1 {\displaystyle x=1} , plug. 1 {\displaystyle 1}

7. Optimization with absolute values

Absolute values can exist in linear optimization problems in two primary instances: in constraints and in the objective function. [2] Absolute Values in Constraints Within constraints, absolute value relations can be transformed into one of the following forms: Where is a linear combination ( where are constants) and is a constant . Form when