How do you graph and solve #abs[ x - 3 ] <=5 #?

2 Answers
Jun 10, 2018

Answer:

# { x | x \in [-2,8], x \in mathbb{R} }#

Explanation:

for # x geq 3 , x-3 geq0 # thus for #x geq 3 , |x-3| = x - 3 #
for # x leq 3 , x-3 leq0 # thus for #x leq 3 , |x-3| = 3-x#

Solve for the two cases:
for # x geq 3, x - 3 \leq 5, #
# x\leq 8 #
thus # 3 \leq x leq 8 #

for # x leq 3, 3-x \leq 5, #
# -2\leq x #
thus # -2 \leq x leq 3#

Find the union of these two intervals:
# [-2,3] \cup [3,8] = [-2,8] #

Thus the solution is # { x | x \in [-2,8], x \in mathbb{R} }#

Graph: enter image source here

Jun 10, 2018

Answer:

#-2 <= x <= 8#

Explanation:

#|x - 3|< = 5#
The simplest way to solve this type of inequality is solving it in 2 separate steps:
a. #(x - 3) <= 5# --> #x <= 8#
b. #-(x - 3) <= 5# --> #- x <= 2#
#x >= - 2#
Answer: #- 2 <= x <= 8#