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
from the definition of absolute value, we know the right side must be ≥0
so x+1 ≥ 0
x ≥ -1
algebraically:
(1/2)x + 1 = x+1 OR (-1/2)x - 1 = x+1
x + 2 = 2x + 2 OR -x - 2 = 2x + 2
x = 0 OR -3x = 4
x = 0 or x = -4/3
which is what you had
remember at the top we said x ≥ -1 , so yes, the -4/3 has to be rejected.
to see this graphically
|1/2x +1| -x-1 = 0
let y = |1/2x +1| -x-1
http://www.wolframalpha.com/input/?i=plot+y+%3D+%7C1%2F2x+%2B1%7C+-x-1
see where it crosses the x-axis ??
only at x = 0
perhaps it will help to view the two functions graphed individually:
http://www.wolframalpha.com/input/?i=+|1%2F2x+%2B1|+%3D+x%2B1
To graph the equation |1/2x + 1| = x + 1, we can follow these steps:
1. Rewrite the equation without the absolute value: 1/2x + 1 = x + 1 (for x ≥ -1) and -(1/2x + 1) = x + 1 (for x < -1).
2. Solve each equation separately for x.
For the first equation, 1/2x + 1 = x + 1:
Subtract x from both sides: -1/2x + 1 = 1
Subtract 1 from both sides: -1/2x = 0
Divide both sides by -1/2: x = 0
For the second equation, -(1/2x + 1) = x + 1:
Distribute the negative sign: -1/2x - 1 = x + 1
Add 1/2x to both sides: -1 = 3/2x + 1
Subtract 1 from both sides: -2 = 3/2x
Multiply both sides by 2/3: x = -4/3
3. Plot the two solutions, x = 0 and x = -4/3, on a graph.
4. Determine the open intervals where the equation is positive or negative.
For x < -1, the equation becomes -(1/2x + 1) = x + 1, so it is positive.
For x > -1, the equation becomes 1/2x + 1 = x + 1, so it is positive.
5. Draw a dashed line for the inequality x > -1 or x < -1, depending on the direction of positivity.
6. Shade the region that satisfies each inequality. For x < -1, shade to the left of the line. For x > -1, shade to the right of the line.
This will give you the graph of the equation |1/2x + 1| = x + 1.