Thus, the solutions are Sometimes absolute value equations have a ridiculous number of cases and it would take too long to go through every single case.Therefore, we can instead graph the absolute value equations using the definition of absolute value as a piecewise function.To get each piece, you must figure out the domain of each piece.
It is equivalent to the distance between and the origin, and is usually called the complex modulus.
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A very basic example would be as follows: if required.
However, these problems are often simplified with a more sophisticated approach like being able to eliminate some of the cases, or graphing the functions.
Absolute value equations are equations involving expressions with the absolute value functions.
This wiki intends to demonstrate and discuss problem solving techniques that let us solve such equations.In this wiki, we intend to discuss this techniques along with strategies on when to use which.is not a possible solution, but it does not mean it's not a possible solution for Case 1 because we're simply going piece by piece in this piecewise function--in the end we will take the union of all possible solutions. how far a number is from zero: "6" is 6 away from zero, and "−6" is also 6 away from zero.So the absolute value of 6 is 6, and the absolute value of −6 is also 6 To show we want the absolute value we put "|" marks either side (called "bars"), like these examples: Rewrite it as: −12 ≤ 3x−6 ≤ 12 Add 6: −6 ≤ 3x ≤ 18 Lastly, multiply by (1/3).If c is a positive number, then | x | = c is equivalent to x = c or x = – c. Break the equation up into two equivalent equations using the rule: If | x | = c then x = c or x = - c . If the absolute values of two expressions are equal, then either the two expressions are equal, or they are opposites.| x – 6 | = 4 is equivalent to x – 6 = 4 or x – 6 = – 4 Step 2. x – 6 6 = 4 6 x = 10 x – 6 6 = – 4 6 x = 2 Step 3 . | 10 – 6 | = | 4 | = 4 | 2 – 6 | = | – 4 | = 4 The solutions are x = 10 and x = 2 . Break the equation up into two equivalent equations using the rule: If | x | = c then x = c or x = - c . If x and y represent algebraic expressions, | x | = Step 1.The absolute value function exists among other contexts as well, including complex numbers.Note that and For complex numbers , the absolute value is defined as , where and are the real and imaginary parts of , respectively.Because we are multiplying by a positive number, the inequalities will not change: −2 ≤ x ≤ 6 Done!Equations with a variable or variables within absolute value bars are known as absolute value equations .