What are absolute value equations?

Answer 1

When dealing with real numbers an absolute value is simply the numerical value (or magnitude) of a real number disregarding the sign. The absolute value of x (denoted |x|) represents the distance that x is from zero on the number line. For example:

|-2| = 2 and |2| = 2 |x| = 3 is translated as "x is 3 units from zero on the number line." Notice, that there are two different numbers that are 3 units away from zero, namely 3 or -3. The solution set of the equation |x| = 3 is {-3,3}, because |-3| = 3 and |3| = 3.

An absolute value equation is an equation with an absolute value. For example: |4x - 2| = 8 is translated that 4x - 2 is 8 units away from zero on the number line. so 4x - 2 = -8 or 4x - 2 = 8 4x = -6 or 4x = 10 x = -3/2 or x= 5/2 Thus, the solution set is { - 3/2, 5/2}. Ex. Solve for x; |x| = -2 This problem has no solution, because the translation is nonsensical. Distance is always positive, distance is not measured in negative values. Ex. Solve for x; |2x - 3| = |3x + 7| This type of equation will be true if either 1. The expression inside the absolute value symbols are exactly the same (that is, they are equal) or 2. The expression inside the absolute values are opposites of each other. 2x - 3 = 3x + 7 or 2x - 3 = - (3x + 7) -3 = x + 10 or 2x - 3 = -3x - 7 -10 = x or 5x = -4 so x = -4/5 The solution set is {-10, -4/5}

Another example

y = |x2 - 8x|

if x = 3 then y = |32 - 8(3)|

y = |9 - 24|

y = |-15|

y = 15 if x = -1 then y = |(-1)2 - 8(-1)|

y = |1 + 8|

y = |9|

y = 9