How to Solve Absolute Value Inequalities

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An absolute value inequality is a type of inequality that contains an absolute value. An absolute value measures the distance a quantity is from 0—for example, |x| measures the distance of x from zero. Absolute value inequalities will find application in symmetries, symmetric limits or boundary conditions. Understand and solve this type of inequality with a few simple steps either by evaluation or transformation. [1]

Steps

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  1. As mentioned above, the absolute value of x, denoted by │𝑥│, is defined as:
    • Absolute value inequalities usually take one of the following forms:
      │𝑥│< or │𝑥│>𝑎 ; │𝑥±𝑎│<𝑏 or │𝑥±𝑎│>𝑏 ; │𝑎𝑥2 +𝑏𝑥│<𝑐
      In this article, the focus will be on inequalities of the form |f(x)|<a or |f(x)|>a , where 𝑓(𝑥) is any function, and a is a constant.
  2. Remember that an absolute value of x can either be positive x or negative x. The absolute value inequality │𝑥│< 3 can also be transformed into two inequalities: -x < 3 or x < 3.
    • For example,│x−3│>5 can be transformed into – (𝑥−3)>5 or 𝑥−3>5.
      │3𝑥+2│ <5 can be transformed into – (3𝑥+2)<5 or 3𝑥+2<5.
    • The term “or” means that either of the two will satisfy the given absolute value problem.
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  3. If it helps, temporarily replace the inequality sign with an equal sign until the end. [2]
  4. Remember that if you divide by a negative number to isolate x on one side of the inequality sign, you will also have to switch the inequality sign. For example, if you divide both sides by -1, -x>5 will become x<-5. [3]
  5. From the values above, you need to write the range of values that can be substituted in for x. This range of values is often referred to as a solution set. Since you would have to solve two inequalities from the absolute value inequality, you will then have two solutions. In the example used above, the solution can be written in two ways: [4]
    1. -7/3<x<1
    2. (-7/3,1)
  6. Pick a number within your solution set, and plug it in for x. If it works, great! If not, go back and check your arithmetic.
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  • Question
    How do I solve 2|3m - 1|-4 = 12?
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    Top Answerer
    First, you want to isolate the absolute value to one side of the equation, so we can add 4 to both sides and get: 2|3m -1| = 16. Then, you would divide both sides by 2 and get: |3m - 1| =8. Here is where the absolute value sign comes in. Because both the positive and the negative value of the number would be the same, you need to solve two different equations. The first one would be: 3m - 1 = 8, and the second one would be: 3m - 1 = -8. So in both equations, you would first have to add 1 on both sides and divide by 3, so we would get m = 3 or m = -7/3. If we plug both of these values back into the equation, you will see that they both work!
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      • The solution set (-3, 3) denotes open interval for both numbers, which means that x can be any number between -3 and 3 excluding both -3 and 3.
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      • For closed intervals, the number on the left side is included as well as on the right side.
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      • The open interval is used for strict inequalities such as x< a or x > a, while the closed interval is used for non-strict inequalities such as x  a or x  a.
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      References

      1. ↑ https://www.khanacademy.org/math/algebra-home/alg-absolute-value/alg-absolute-value-inequalities/v/absolute-value-inequalities
      2. ↑ https://www.chilimath.com/lessons/intermediate-algebra/absolute-value-inequalities/
      3. ↑ https://www.chilimath.com/lessons/intermediate-algebra/absolute-value-inequalities/
      4. ↑ http://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueIneq.aspx
      5. Leithold, Louis. The Calculus with Analytic Geometry, 6th ed. pp. 2-10.

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