Question 804739: What is |2x+5|=23
Found 2 solutions by Alan3354, DrBeeee:
Answer by Alan3354(69443)   (Show Source):
Answer by DrBeeee(684)  (Show Source):
You can put this solution on YOUR website! This is tricky, so I'm going to show you how I handle the absolute value in order to solve this equation analytically.
Before we solve the problem, let's learn about the absolute value (later when you take complex variables you will work with a simular function called the absolute magnitude).
Here's the trick. Since we can't mathematically manipulate |z|, we replace it with a term that we can use. Note that when
(1) z >= 0, z is already positive, so we can replace |z| with z or
(2) |z| = z when z>=0, OK?
On the other hand, when
(3) z <= o, z is negative, so we must use -z to get a positive value or
(4) |z| = -z when z<=0. Get it?
Now let's apply this substitution to the given problem
(5) |2x+5| = 23
When (2x+5) is positive, greater or equal to zero, we have
(6) 2x + 5 >= 0 or
(7) 2x >= -5 or
(8) x >= -5/2
In this case, we let |2x+5| = +(2x + 5), and (5) becomes
(9) 2x+5 = 23 or
(10) 2x = 23 - 5 or
(11) 2x = 18 or
(12) x = 9 since x>=-5/2
On the flip side, when (2x+5) is negative, less than or equal to zero, we have
(6) 2x + 5 <= 0 or
(7) 2x <= -5 or
(8) x <= -5/2
In this case, we let |2x+5| = -(2x + 5), and (5) becomes
(9) -(2x+5) = 23 or
(10) 2x = -23 - 5 or
(11) 2x = -28 or
(12) x = -14 since x<=-5/2
Your answer is x = {-14,9}
If you graph the equation (5), note that it is centered on x = -5/2 and y = -18, and the solution points are (-14,0) and (9,0). Also note that the "distance" between the solutions {-14,9} is 23, and -5/2 is half way between the solution points.
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