How do you graph and solve #5 + |1-x/2| >=8#?
1 Answer
Explanation:
1) Simplifying
First of all, bring
#5 + abs(1 - x/2) >= 8#
#<=> abs(1 - x/2) >= 3 #
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2) Evaluating the absolute value function
To evaluate the absolute value function, we need to find out when
To do this, let's find the point where
# 1 - x/2 = 0 " " <=> " " x/2 = 1 " " <=> " " x = 2#
Plugging
# 1 - x/2 >= 0 color(white)(xxx) "for " x <= 2#
# 1 - x/2 < 0 color(white)(xxx) "for " x > 2#
Now, you can evaluate the absolute value function:
# abs(1 - x/2) = { (color(white)(xx) 1 - x/2, color(white)(xxx) "for " 1 - x/2 >= 0 ), (-(1 - x/2), color(white)(xxx) "for " 1 - x/2 < 0) :}#
# color(white)(xxxxx) = { (color(white)(x) 1 - x/2, color(white)(xxxxx) "for " x <= 2 ), (-1 + x/2, color(white)(xxxxx) "for " x > 2) :}#
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3) Solving the two cases
3a) Let
This means that
# => 1 - x/2 >= 3#
... subtract
#<=> - x/2 >= 2#
... multiply both sides with
Be careful: if multiplying with a negative number or dividing by a negative number, you need to flip the inequality sign!
#<=> x <= - 4#
Now, we need to combine the condition
As
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3b) Let
This means that
# => - 1 + x/2 >= 3#
... add
# <=> x/2 >= 4#
... multiply both sides of the inequality with
#<=> x >= 8#
Between the two conditions,
Thus, this is the solution for the second case.
In total, the solution is
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4) Graphing
You can graph the absolute value function
- the "elbow" is the point of the function where
#1 - x/2 = 0# holds which is#x = 2# . Thus, the elbow is#(2; 0)# . - The slope is the factor of
#x/2# , so it's#1/2# .
Thus the absolute function looks as follows:
graph{abs(1 - x/2) [-10, 10, -5, 5]}
The graph of
#abs(1 - x/2) >= 3#
is the part of the graph that is above the horizontal line at
graph{(y - abs(1 - x/2))(y - 3) = 0 [-15, 15, -5, 10]}
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Hope that this helped! :-)