We are asked for the similarities and differences of solving the ∣2x+1∣=∣x−7∣ and the 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.
Equation
To solve the equation
4x+3=-2x+9, we will create two out of the left- and right-hand sides of the equation.
y=4x+3 and y=-2x+9
The
x- where the graphs of these functions is the solution to our equation.
The graphs intersect at x=1, which is our solution.
Absolute Value Equation
An absolute value equation in the form
∣ax+b∣=∣cx+d∣ has two related equations.
Equation 1:ax+bEquation 2:ax+b=-(cx+d=-(cx+d)
Let's write the two related equations for
∣2x+1∣=∣x−7∣.
Equation 1:2x+1Equation 2:2x+1=-(x−7=-(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+1and 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).
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2x+1=-(x−7)
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First Function
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y=2x+1
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Second Function
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y=-(x−7)⇔y=-x+7
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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.
Similarities and Differences
We can now state the similarities and differences between these two processes.
- Similarities: When solving the absolute value equation, we write two equivalent equations and solve them in the same way that we solve 4x+3=-2x+9. That is, by writing and graphing a for each side of the equation. The x-coordinate where the graph of these equations intersects is the solution to the original equation.
- Differences: The absolute value equation has two solutions, one for each formed by its related equations. Conversely, 4x+3=-2x+9 has only one solution.