Question 996966: absolute value inequalities.
k-7+6< 8
i tried working out the problem with both positive 8 and negative 8.
and i got k<9 and k<-7 is this an and or an or problem? my teacher keeps telling me im getting it wrong and i dont understand it.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! first thing you need to do is have the absolute value portion of the equation on the left side of the equal sign and everything else on the right side of the equal sign.
|k - 7| + 6 < 8
subtract 6 from both sides of the equation to get:
|k - 7| < 8 - 6
simplify to get:
|k - 7| < 2
now if the expresion of x - 7 is positive, then you remove the absolute value sign and solve as is.
you get:
(x - 7) < 2
remove parentheses to get:
x - 7 < 2
add 7 to both sides of the equation to get:
x < 9
if the expression nside the absolute value sign is negative, then you remove the absolute value sign and you revgerse the inequality and you change the sign of the side of the equation that did not include the absolute value expression.
you will get:
|x - 7| < 2 becomes:
(x - 7) > - 2
the inequality was reversed and the right side of the equation was made negative.
remove the parnetheses to get:
x - 7 > - 2
add 7 to both sides of the equation to get:
x > 5
your solution is that the original equation is true when x < 9 and when x > 5.
your original equation is |x - 7| + 6 < 8
evaluate this equation for when x = from 4 to 10.
when x = 4, you get 9 < 8 which is false.
when x = 5, you get 8 < 8 which is false (it's equal but not less than)
whebn x = 6, you get 7 < 8 which is true.
when x = 7, you get 6 < 8 which is true.
when x = 8, you get 7 < 8 which is true.
when x = 9, you get 8 < 8 which is false (it's equal but not less than)
when x = 10, you get 9 < 8 which is false.
the equation is only true when x < 9 and x > 5.
those values are x = 6,7,8.
here's a reference on how to solve absolute value inequalities.
http://www.regentsprep.org/regents/math/algtrig/ate2/absinequal.htm
in case you didn't get through that one real well, here's a reference on how to solve absolute value equations.
http://www.regentsprep.org/regents/math/algtrig/ATE1/abslesson.htm
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