Absolute Value Rules

Question 1

When two absolute values are equal (i.e.)

|16x + 14| = |8x + 6|, how do we solve?

Answer 1

The expressions within the absolute value bars are either equals or opposites.

Either:
16x + 14 = 8x + 6
OR
16x + 15 = -(8x+6)

Question 2

What is the rule of adding absolute values?

Answer 2

|a+b| <= |a| + |b|

Question 3

When |a+b| = |a| + |b|

Answer 3

either a/b or both a and b are 0 OR both a AND b share the same sign

Question 4

What is the rule of subtracting absolute values?

Answer 4

|a-b| >= |a| - |b|

Question 5

When |a-b| = |a| - |b|

Answer 5

either b is 0 OR a and b have the same sign

Question 6

If an absolute value equation has a variable on both sides of the equation

Answer 6

we solve as usual BUT we must check the solutions of the equation for extraneous solutions.

Question 7

If the absolute value of an expression is equal to a negative number, i.e. (|2x+ 3| = -2)

Answer 7

there are no solutions to that equation

Question 8

If |a-b| = |a| - |b|

Answer 8

then a and b must have the same sign and |a| >= |b|

Question 9

When |a+b| = |a| + |b| then

Answer 9

one or both quantities are 0 OR a and b must have the same sign

Question 10

For inequalities with variables . . . .

Answer 10

never multiply or divide an inequality by a variable unless the variable’s sign is known. Otherwise, we don’t know whether to flip the sign or not.