Absolute Values and Inequalities

Question 1

how do you solve questions like:
For what values of x does |2x+8|= |4x-4|?

Answer 1

the equation is equal when the insides are either both positive or both negative and when they are opposite (one pos, one neg, and the other pos, the other neg). No need to test all four though, just need to solve once for same sign and once for opposite signs (2x+8=4x-4, and 2x+8=-4x+4)

Question 2

|a+b| ? (<, >=, etc.) |a|+|b|

Answer 2

<=

Question 3

|a-b| ? (<, >=, etc.) |a|-|b|

Answer 3

> = (since if b is negative |a-b| results in a+b

Question 4

if b!=0 and |a-b| = |a| - |b|, then…?

Answer 4

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

Question 5

how many solutions are there to the equation:
|x+1| = -2

Answer 5

0, since there is no value for x that can yield a negative value for |x+1| since absolute value output is always positive

Question 6

when must you check solutions to an absolute value equation to be sure the solutions ecist

Answer 6

when there are variables on both sides of the equation like |x+1|=2x, since you dont know whether x is negative

Question 7

if |a| = |b|, what must be true about a, and b?

Answer 7

they are equal or have opposite signs

Question 8

what must be true of a, and b if |a+b|=|a|+|b|?

Answer 8

a, or b are zero, or they are both of the same sign

Question 9

can you add inequalities?

Answer 9

yes, as long as they are facing the same way. this would help with questions like: what can we say about x+y given that x<26 and y<8? adding the two yields x+y < 34

Question 10

T/F: you never divide or multiply a variable in an inequality if you do not know its sign

Answer 10

true

Question 11

are there any solutions that satisfy an equation like this: |x+3|=-2?

Answer 11

no, since an abs val cant equal a negative