I am asked to solve the absolute value equation of
|1/2x +1| =x+1 graphically. I can solve this algebraically and consider two cases for the absolute value one being positive and the other negative. I know this will be a straight line and when I solved it algebraically x=0 the other solution is -4/3 which is an extraneous root and does not work. How do I graph this? Thanks for your help
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To graph the equation |1/2x + 1| = x + 1, we can follow these steps:
1. Rewrite the equation without absolute value:
- When 1/2x + 1 is positive: 1/2x + 1 = x + 1
- When 1/2x + 1 is negative: -(1/2x + 1) = x + 1
2. Simplify both equations:
- For the positive case: 1/2x + 1 = x + 1
Simplifying gives us x = 0.
- For the negative case: -(1/2x + 1) = x + 1
Distribute the negative sign and simplify to get -1/2x - 1 = x + 1
Combine like terms: -1 - 1 = x + 1/2x
Simplify further: -2 = 3/2x
Divide both sides by 3/2: x = -4/3
3. Plot the solutions on a graph:
- Plot x = 0, which is a vertical line passing through the point (0,0).
- Plot x = -4/3, which is another vertical line passing through the point (-4/3,0).
Note: The graph consists of two vertical lines intersecting at the point (-4/3, 0).
Remember to always check the extraneous roots to see if they satisfy the original equation. In this case, x = -4/3 is an extraneous root because it does not satisfy the absolute value equation.