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Question 935768: Find the domain of the inequality √(x+3) ≤ x+1..Draw the graph to find the the common point(if any).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the domain is all x greater than or equal to -3.
this is because when x is smaller than -3, you get the square root of a negative number which is not real.
the equation will be equal when x is equal to 1.
the graph of the two equations is:
y = sqrt(x+3)
y = x + 1
you can see that the graph of sqrt(x+3) is smaller than or equal to the graph of x + 1 when x is greater than or equal to 1.
if you subtract one equation from the other, you will get:
sqrt(x+3) <= (x+1) becomes:
sqrt(x+3) - (x+1) <= 0
the inequality will be true when sqrt(x+3) - (x+1) <= 0
to graph that equation, set it equal to y to get:
y = sqrt(x+3)^2 - (x+1)
that graph is shown below:
you can see that the equation becomes less than or equal to 0 when x is greater than or equal to 1.
you can solve for the equal points as shown below:
start with:
sqrt(x+3) <= (x+1)
square both sides to get:
(x+3) <= (x+1)^2
simplify to get:
(x+3) <= x^2 + 2x + 1
subtract (x+3) from both side to get:
0 <= x^2 + 2x + 1 - (x+3)
simplify to get:
0 <= x^2 + 2x + 1 - x - 3
simplify further to get:
0 <= x^2 + x - 2
factor the equality portion of this equation to get:
0 = (x+2) * (x-1)
solve for x to get:
x = -2 or x = 1
replace x in the original equation to confirm the solution is good.
when x = 1:
sqrt(x+3) <= x+1 becomes sqrt(4) <= 2 which becomes 2 <= 2 which is true.
when x = -2:
sqrt(x+3) <= x+1 becomes sqrt(1) <= -2+1 which becomes 1 <= -1 which is false.
only x = 1 is a good solution.
x = -2 is an extraneous solution that cannot be confirmed in the original equation and is therefore not a solution.
the inequality is true when x is greater than or equal to 1.
confirm this by checking your inequality in the regions of -3 <= x <= 1 and x > 1.
since you graphed this, you don't need to, but you can confirm anyway just to make sure you didn't mess anything up.
when x = -3, the inequality becomes 0 <= -2 which is false.
when x = -2, the inequality becomes 1 <= -1 which is false.
when x = -1, the inequality becomes sqrt(2) <= 0 which is false.
when x = 0, the inequality becomes sqrt(3) <= 1 which is false.
wen x = 1, the inequality becomes sqrt(4) <= 2 which is true.
when x = 2, the inequality becomes sqrt(5) <= 3 which is true.
when x = 3, the inequality becomes sqrt(6) <= 4 which is true.
the calculations confirm the graph.
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