1-5 Absolute Value Equations and Inequalities
Recall: What is absolute value?
-It is the distance a number is from zero
 That’s why it’s never negative; distance can’t be negative
Def: Algebraic Definition of Absolute Value

If
x0

If
x0
then x  x
then x   x
o Note: “-x” does NOT indicate “negative x”, rather it indicates “the
opposite of x”
o Ex. If x=6 then |6|=6; If x=-6 then |-6|=-(-6)=6
Solving Absolute Value Equations
1. Isolate the absolute value expression
2. Rewrite the equation as 2 separate equations
a. 1st equation: Drop the absolute value bars and solve the equation
b. 2nd equation: Drop the absolute value bars, negate the opposite side, and solve
3. Check your solutions
Ex. Solve.
1. x  8  3
2. y  4  3  0  isolate before rewriting the 2 equations
3. 8  5a  14  a  make sure the negative is applied to both the 14 and a
4. 6 5x  2  312  isolate before rewrite
5. 3d  9  6  0  isolate first; no need to go further since abs. value can’t be negative
6. 3 x  2  7  14  isolate before rewrite
7. 2 x  5  3x  4  only 1 real solution since other is extraneous
Def: extraneous solution – a solution of an equation derived from an original equation that
is not a solution of the original equation
1-5 Absolute Value Equations and Inequalities
Absolute Value Inequalities
Let k represent a positive real number.
a.
x  k is equivalent to x  k or x  k
b.
x  k is equivalent to k  x  k
Solving Absolute Value Inequalities
1. Isolate the absolute value expression
2. Rewrite as a compound inequality
c. 1st equation: Drop the absolute value bars and solve the inequality
d. 2nd equation: Drop the absolute value bars, negate the opposite side, and
reverse the inequality sign
Ex. Solve and graph the solution.
1. 2 x  4  12
2. 2 4  x  10
3. 3 2 x  6  9  15