Absolute value is the modulus of a real number or of a complex number. For example, the absolute value of -2.3 written |-2.3| is 2.3. The absolute value of a complex number is also the modulus, for example, the absolute value of \(2 + 3i\) is \(\sqrt{2^2 + 3^2}\).
Absolute Value Calculator
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Absolute Value Equation
The absolute value equations are equation where the variable is within an absolute value operator like \(|x + 2| = 6\). The challenge is that the absolute value of a number depends on the number sign: If it is positive, then it’s equal to the number: \(|6| = 6\). I f the number is negative it will be \(|-6| = 6\).
Examples
1- \( |x| = 9 \)
\( x = \pm 9 \)
\( |9| = 9 \) and \( |-9| = 9 \)
2- \( |2x – 3| = 9 \)
\( 2x – 3 = 9 \) and \( 2x – 3 = -9 \)
\( x = 6 \) and \( x = 3 \)
Note: The absolute value of 8 is 8, so first solution does work. Similarly, the another one also.
3- \( |x + 3| = 4 \)
Now notice that there is no solution of this particular problem. Just because of value have negative sign after the sign of equality.
4- \( |x + 3| = |x – 11| \)
\( x + 3 = x – 11 \)
\( 3 = – 11 \) No solution for this portion of solution
5- \( |x – 5| = 10 \)
\( x = 15 \) and \( x = -5 \)
Notice here both these number are exactly 10 away from number +5.
6- \( |x + 2| = 6 \)
\( x + 2 = 6 \) and \( x + 2 = -6 \)
\( x = 4 \) and \( x = -8 \)
Notice here both of these numbers are 6 away from -2.
Graph
\( y = |x| \)
| X | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| Y | 2 | 1 | 0 | 1 | 2 |
Graph of Equation
1- \( y = |x + 2| \)
Here slope is 1.
2- \( y = |x – 3| \)
FAQS
Q1: What is a fact about absolute value?
Answer: It is the magnitude of a number (real or complex) graphically the absolute value represents displacement from the 0 and therefore it is a non-negative.
Q2: Why does we can’t solve \(|x| = -1\)?
Answer: The absolute value of a number is its displacement or distance away from origin (zero) number will be positive as we can’t show distance with a negative sign.
Q3: Why do we use absolute value?
Answer: Normally, we use in problem solving where distance are involved and some time with inequalities.
Q4: What are the practical uses of absolute value in our daily life?
Answer: Whenever scientist discuss about energy waves, there are both negative and positive direction of movement, it can also be used to define the difference of changes from one point to another.
Q5: How to solve absolute value?
Answer: To solve an equation where absolute value exist, keep the absolute value on one side that is known as isolated and then set its content equation to both the positive and negative value of the number, on the other side of the equation and solve both sides of the equation.
Q6: Can the absolute value of a number ever be 0?
Answer: The definition of a number absolute value is its distance from 0. Since 0 is zero units away from itself, thus absolute value of a zero is just 0
Q7: What are the properties of absolute value?
Answer:
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Non-negative \(|a| ≥ 0\).
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Positive- definiteness \( |a| = 0 \) and \( a = 0 \)
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Multiplicatively \( |ab| = |a| |b| \)
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Sub- additivity \( |a + b| ≤ |a| + |b| \)
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Idempotentance \(||a|| = |a|\)
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Symmetry \(|-a| = |a|\)
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Triangle inequality \( |a – b| ≤ |a – c| + |c – b| \)
Q8: What is the specialty of absolute value?
Answer: The absolute values of x is always either positive or zero but never negative.
Q9: What is the absolute value?
Answer: It is the distance of a number from 0.
Q10: What is an absolute value equation?
Answer: Equation where the variable is within an absolute value operator, for example \(x – 6 = 7\)
Q11: What are the steps in solving absolute value equation?
Answer:
Step 1: Isolate the absolute value.
Step 2: Solve for positive and solve for negative value.
Step 3: Verify or check the answer.