How to Solve Double Absolute Value Inequality?
Introduction
In mathematics, generalisations of the absolute value for real numbers occur in a wide variety of mathematical analysis. They are used to define complex numbers, quaternians, ordered rings and vector spaces. In this article, you will learn to solve double absolute values inequaliteis step-by-step.
Solution Method of Double Absolute Value Inequalities
There are some steps to solving double absolute value inequalities. We can solve any problem related to double absolute values by following these steps.
- Isolate the absolute value inequality expression on both sides.
- Break the expression into two inequalities concerning left absolute value.
- Again isolate both expressions and then breaking into two further expressions.
- Find the solution for the variable involved and graph the solution.
See the below examples to find the solution to double absolute value inequalities. If you want to solve it quickly so you can also use online calculator.
Double Absolute value inequality example no.1
Solve |x - 3|- 6 ≤ x - |x + 2|
To solve the given example, we will use the above mentioned steps therefore,
Step-I
Isolating the equation on both sides by adding 6 on both sides we get,
|x - 3| ≤ x + 6 - |x + 2|
Step-II
Breaking the expression into 2 pieces, we get
Step-III
| Case 1: x - 3 ≤ x + 6 - |x + 2| |
Case 2: x - 3 ≥ - x - 6 + |x + 2| |
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Isolating by adding – x - 6 both sides.
-9 ≤ - |x + 2|
More simplification to remove negative sign.
9 ≥ |x + 2|
Or,
|x + 2| ≤ 9
x -3 ≥ -x -6 + |x + 2|
Isolating by adding x + 6 both sides.
2x + 3 ≥ |x + 2|
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Isolating by adding x + 6 both sides.
2x + 3 ≥ |x + 2|
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Now breaking these expressions for further expression by using inequality signs.
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x + 2 ≤ 9 and x + 2 ≥ - 9
And
x ≤ 7 and x ≥ -b11
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x + 2 ≤ 2x + 3 and x + 2≥ - 2x - 3
And
x ≤ -1 and x ≥ (-5) / 3
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Step-IV
Solutions from the expression are
x ≤ 7 and x ≥ -11
x ≤ -1 and x ≥ (-5) / 3
By comparing the solution it is obvious that the solution overlap with the values
x ≤ 7 and x ≤ - 1
Double Absolute value inequality example no.2
Solve |7x - 3| ≥ |3x + 7|
Step-I
We don’t need to convert the given equation into isolated form because it is already in it. So we will move to the next step and we can solve it by an alternative method.
Step-II
Breaking the equation into two equations as
7x - 3 ≤ - (3x + 7) and 7x - 3 ≥ 3x + 7
7x - 3 ≤ - 3x - 7 and 7x - 3 ≥ 3x + 7
Step-III
Simplifying both
10x ≤ -4 and 4 x ≥ 10
Dividing by 4 both sides.
x ≤ (-4) / 10 and x ≥ (10) / 4
x ≤ (-2 ) / 5 and x ≥ (5) / 2
Step-IV
Hence the solution is
x ≤ (-2) / 5 and x ≥ (5) / 2
FAQ’s
What does double absolute value mean in math?
Double absolute value means that an expression has absolute value or modulus (||x||) two times. It can be either on one side of the expression or both sides. For example,
||ax+b|+c|
How do you Solve Double Absolute Value?
You can solve double absolute value inequalities by following the given steps
- Isolate the expression on both sides of absolute value.
- Breaking the expression into two further expressions.
- Again isolating the two expressions.
- Finding the solution for x and graphing it.
Where Do We Use Absolute Value Inequalities?
These inequalities are used in different fields of mathematics, such as to define complex number, vector space or ordered rings. They have also many real life uses. One of most important use is in margin of error or tolerance.
What are the Rules for Absolute Value?
The rules of for absolute value are
1. |– a| = |a|
2. |a| ≥ 0
3. Products |ab| = |a| |b|
4. Quotients |a / b| = |a| / |b|
5. Powers |a ^ n| = 〖|a|〗^ n
6. Triangle Inequality |a + b| ≤ |a| + |b|
7. Alternate Triangle Inequality |a – b| ≥ |a| – |b|