SOLUTION: |x|+|x+2|>= 1

Question 268651: |x|+|x+2|>= 1
Answer by persian52(161) (Show Source): You can put this solution on YOUR website!
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|x|+|x+2|>=1
Since |x+2| does not contain the variable to solve for, move it to the right-hand side of the inequality by subtracting |x+2| from both sides.
|x|>=-|x+2|+1
Find the point where each absolute value expression goes from negative to positive. To accomplish this, set the expressions inside each absolute value equal to 0.
x=0_x+2=0
Solve the inequality for x to find the point where x+2 goes from negative to positive.
x=-2
Evaluate each of the absolute value expressions over the interval x<=-2. If the absolute value expression is negative over the interval, replace the absolute value with the negative portion of the absolute value. If it is positive over the interval, just remove the absolute value.
(-(x))>=-(-(x+2))+1
Evaluate each of the absolute value expressions over the interval -2 (-(x))>=-(x+2)+1
Evaluate each of the absolute value expressions over the interval x>0. If the absolute value expression is negative over the interval, replace the absolute value with the negative portion of the absolute value. If it is positive over the interval, just remove the absolute value.
(x)>=-(x+2)+1
Solve each of the equations to find the complete solution set.
(-(x))>=-(-(x+2))+1 where x<=-2_(-(x))>=-(x+2)+1 where -2=-(x+2)+1 where x>0
Remove the parentheses around the expression -(x).
-(x)>=-(-(x+2))+1
Multiply -1 by each term inside the parentheses.
-(x)>=-((-x-2))+1
Remove the parentheses around the expression -x-2.
-(x)>=-(-x-2)+1
Multiply -1 by each term inside the parentheses.
-(x)>=(x+2)+1
Add 1 to 2 to get 3.
-(x)>=x+3
Multiply -1 by each term inside the parentheses.
-x>=x+3
Move all terms not containing x to the right-hand side of the inequality.
-2x>=3
Solve the first equation for x.
x<=-(3)/(2)
To set the left-hand side of the inequality equal to 0, move all the expressions to the left-hand side.
(-(x))+(x+2)-1>=0
Remove the parentheses that are not needed from the expression.
-(x)+x+2-1>=0
Multiply -1 by each term inside the parentheses.
-x+x+2-1>=0
Combine all similar terms in the polynomial -x+x+2-1.
1>=0
Since 1=0, the equation will always be true.
Always True
Solve the second equation for x.
No Solution
Remove the parentheses around the expression x.
x>=-(x+2)+1
Multiply -1 by each term inside the parentheses.
x>=(-x-2)+1
Add 1 to -2 to get -1.
x>=-x-1
Move all terms not containing x to the right-hand side of the inequality.
2x>=-1
Solve the third equation for x.
x>=-(1)/(2)
To find the final solution set, verify the solutions fall into the defined interval for each equation. x<=-(3)/(2) does not fall within the defined interval x<=-2 for that portion of the solution, therefore it is not a valid solution. x>=-(1)/(2) does not fall within the defined interval x>0 for that portion of the solution, therefore it is not a valid solution.
No Solution
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