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Consider that an absolute value equation in the form |ax+b|=|cx+d| has two related equations.
See solution.
We are asked for the similarities and differences of solving the absolute value equation |2x+1|=|x-7| and the equation 4x+3=-2x+9 by graphing. To do so, we will first go through the whole process of solving the equation and the absolute value equation. Then, we will find the similarities and differences between these processes.
To solve the equation 4x+3=-2x+9, we will create two functions out of the left- and right-hand sides of the equation. y=4x+3 and y=-2x+9 The x-coordinate where the graphs of these functions intersect is the solution to our equation.
The graphs intersect at x=1, which is our solution.
An absolute value equation in the form |ax+b|=|cx+d| has two related equations. Equation1: ax+b&= cx+d Equation2: ax+b&=-(cx+d) Let's write the two related equations for |2x+1|=|x-7|. Equation1: 2x+1&= x-7 Equation2: 2x+1&=-(x-7) This means that for the absolute value equation, we have to solve the two related equations in the same way as we have solved 4x+3=-2x+9. Let's create the two functions out of the left- and right-hand sides for 2x+1=x-7. y=2x+1 and y=x-7 Again, the x-coordinate where the graphs of these equations intersect is a solution for the absolute value equation.
The graphs intersect at x=-8, which is our solution. Now, let's create the two functions out of the left- and right-hand sides for 2x+1=-(x-7).
| 2x+1=- (x-7) | |
|---|---|
| First Function | y=2x+1 |
| Second Function | y=- (x-7) ⇔ y=- x+7 |
Once more, the x-coordinate where the graphs of these functions intersect is the solution to our equation.
The graphs intersect at x=2, which is our solution. Therefore, for the absolute value equation, we have two solutions, x=-8 and x=2.
We can now state the similarities and differences between these two processes.