Question 902237
start with 7 - abs(2x+3) <= 6


subtract 7 from both sides of the equation to get:


- abs(2x+3) <= -1


multiply both sides of this equation by -1 to get:


abs(2x+3) >= 1


when 2x+3 >= 0, this equation becomes:


2x+3 >= 1


solve for x to get x >= -1


when 2x+3 < 0, this equation becomes:


2x+3 <= -1


solve for x to get x <= -2


your solutions are:


x >= -1 or x <= -2


in interval notation this solution is:


(-infinity,-2] or [-1,infinity)


you would graph this equation as follows:


start with abs(2x+3) >= 1


graph y = abs(2x+3) and graph y = 1 on the same graph.


your solution lies in the region where the graph of y = abs(2x+3) is greater than or equal to the graph of y = 1.


that occurs when x <= -2 and when x >= -1.


that graph is shown below:


<img src = "http://theo.x10hosting.com/2014/091603.jpg" alt="$$$" </>


you can confirm the solution is good by going back to the original equation and replacing x with selected values to see when the equation is true and when the equation is false.


the original equation is 7 - abs(2x+3) <= 6


when x is -3, you get 7 - abs(-6+3) = 7 - abs(-3) = 7 - 3 = 4 which is smaller than or equal to 6 so all is good because x is smaller than or equal to -2.


when x is -2, you get 7 - abs(-4+3) = 7 - abs(-1) = 7 - 1 = 6 which is smaller than or equal to 6 so all is still good because x is smaller than or equal to -2.


when x ix 0, you get 7 - abs(0+3) = 7 - abs(3) = 7 - 3 = 4 which is smaller than or equal to 6 so all is still good because x is greater than or equal to -1.


when x is -1, you get 7 - abs(-2+3) = 7 - abs(1) = 7 - 1 = 6 which is smaller than or equal to 6 so all is still good because x is greater than or equal to -1.


when x is -1.5, you get 7 - abs(-3+3) = 7 - abs(0) = 7 - 0 = 7 which is greater than 6 so all is still good because x is not greater than or equal to -1 and x is not smaller than or equal to -2 and you should not get a result smaller than or equal to 6 which you don't.


everything looks good, so your solution is:


x <= -2 or x >- -1


interval notation is as shown above and below:


(-infinity,-2] or [-1,infinity)