How do you graph and solve #|2x+3| <= 7#?
1 Answer
Explanation:
Graphing
Let's start with graphing.
You can graph
The graph of
graph{|x| [-15, 15, -3, 12]}
To graph
- the slope is being described by the factor in front of
#x# . Here, the slope is#2# . - to find the "elbow", set
#2x + 3 = 0# and solve for#x# . Thus, the "elbow" is at#(-3/2, 0)#
Thus, the graph of
graph{|2x+3| [-15, 15, -3, 12]}
Now, the solution of the equation are all
graph{(y - |2x+3|)(y-7) = 0 [-15, 15, -3, 12]}
So, you already see on the graph that the solution is
How to find the same solution without the graph?
Solving
To solve the inequality, you need to evaluate the absolute value
Generally, for an absolute value, you have
In this case, you need to know for which
We already know that this is the case for
So, we have
Thus, we need to consider the two cases:
1)
#2x + 3 <= 7#
#<=> x <= 2 # Don't forget to take a look at the condition
#x >= -3/2# which needs to hold at the same time.
Here, the solution is#-3/2 <= x <= 2# or#x in [-3/2; 2]#
2)
#-(2x + 3) <= 7#
#<=> -2x - 3 <= 7 #
#<=> -2x <= 10 # ... divide by
#-2# and don't forget to flip the inequality sign (this needs to be done if multiplying with or dividing by a negative number)...
#<=> x >= -5# Don't forget to take a look at the condition
#x < -3/2# which needs to hold at the same time.
Thus, the solution for this case is#-5 <= x < -3/2# or#x in [-5; -3/2)#
In total, we have
