Absolute value

Question 1

absolute value (definition)

Answer 1

|x|= {x, x ≥ 0; –x, x < 0

Question 2

absolute value product rule

Answer 2

|ab|=|a||b|

Question 3

absolute value power rule

Answer 3

|aⁿ|=|a|ⁿ, whenever aⁿ is defined

Question 4

absolute value quotient rule

Answer 4

|a/b|=|a|/|b|

Question 5

equality properties of absolute value

Answer 5

1. |x|= 0 if and only if x = 0
2. for c > 0, |x|= c if and only if x = c or –x = c
3. for c < 0, |x|= c has no solution
4. a number and its negative have the same absolute value: |x| = |–x|

Question 6

the triangle inequality

Answer 6

for real numbers a and b,

|a + b| ≤ |a| + |b|

Question 7

for a real number c > 0, |x| < c

Answer 7

is equivalent to –c < x < c

Question 8

for a real number c > 0, |x| ≤ c

Answer 8

is equivalent to –c ≤ x ≤ c

Question 9

for a real number c ≤ 0, |x| < c

Answer 9

has no solution

Question 10

for a real number c < 0, |x| ≤ c

Answer 10

has no solution

Question 11

for a real number c ≥ 0, |x| > c

Answer 11

is equivalent to x < –c or x > c

Question 12

for a real number c ≥ 0, |x| ≥ c

Answer 12

is equivalent to x ≤ –c or x ≥ c

Question 13

for a real number c < 0, |x| ≥ c

Answer 13

is true for all real numbers

Question 14

for a real number c < 0, |x| > c

Answer 14

is true for all real numbers