Pre-Calculus 11 Absolute Value Equations

Lesson Focus: To solve an absolute value equation graphically, with or without technology; to algebraically

solve an equation with a single absolute value and verifying the solution; to explaining why the absolute value

equation |f(x)| = b for b < 0 has no solution.

an absolute value equation is an equation that includes the absolute value of an expression involving a

variable

i.e. 62 x

to solve an absolute value equation graphically:

1. graph the left side and the right side of the equation on the same set of axes

2. the point(s) of intersection are the solution(s)

e.g. Solve the following absolute value equations graphically.

1. 53 x 2. 26 x

Y1 = Y1 =

Y2 = Y2 =

to solve an absolute value equation by algebraically:

1. consider to separate cases

a) Case 1: The expression inside the absolute value symbol is greater than or equal to zero

b) Case 2: The expression inside the absolute value symbol is less than zero

2. we can consider both cases by replacing the absolute value brackets with a plus/minus symbols

i.e. becomes

3. the roots in each case are the solution

4. there may be extraneous roots that need to be identified and rejected

we can verify solutions by substituting into the original equation

e.g. Solve the following absolute value equations algebraically. Check for extraneous roots.

1. 83 x 2. 5423 xx

since, by definition, absolute value is greater than or equal to zero, there can be no solution if axf ,

where a < 0

the empty set is a set with no elements and is symbolized by {} or ∅

e.g. Solve 3715 x .

an absolute value equation can have two solutions and a quadratic equation can also have two solutions

therefore an absolute value equation involving a quadratic solution can have four solutions

if we cannot factor the quadratic expression, remember to use the quadratic formula

a

acbbx

2

42

e.g. Solve algebraically.

1. 862 xx 2. xx 12

e.g. A computerized process controls the amount of fish that is packaged in a specific size of can. The

computer program sets the ideal mass at 170 g but allows a tolerance of 6 g. Solve an absolute value

equation for the maximum and minimum mass, m, of fish in this size of can.