document.write( "Question 935768: Find the domain of the inequality √(x+3) ≤ x+1..Draw the graph to find the the common point(if any). \n" ); document.write( "
Algebra.Com's Answer #569118 by Theo(13342)\"\" \"About 
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the domain is all x greater than or equal to -3.
\n" ); document.write( "this is because when x is smaller than -3, you get the square root of a negative number which is not real.
\n" ); document.write( "the equation will be equal when x is equal to 1.
\n" ); document.write( "the graph of the two equations is:
\n" ); document.write( "y = sqrt(x+3)
\n" ); document.write( "y = x + 1
\n" ); document.write( "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.
\n" ); document.write( "\"+graph%28600%2C600%2C-5%2C5%2C-10%2C10%2Csqrt%28x%2B3%29%2Cx%2B1%29\"
\n" ); document.write( "if you subtract one equation from the other, you will get:
\n" ); document.write( "sqrt(x+3) <= (x+1) becomes:
\n" ); document.write( "sqrt(x+3) - (x+1) <= 0
\n" ); document.write( "the inequality will be true when sqrt(x+3) - (x+1) <= 0
\n" ); document.write( "to graph that equation, set it equal to y to get:
\n" ); document.write( "y = sqrt(x+3)^2 - (x+1)
\n" ); document.write( "that graph is shown below:
\n" ); document.write( "you can see that the equation becomes less than or equal to 0 when x is greater than or equal to 1.
\n" ); document.write( "\"+graph%28600%2C600%2C-5%2C5%2C-10%2C10%2Csqrt%28x%2B3%29-%28x%2B1%29%29\"
\n" ); document.write( "you can solve for the equal points as shown below:
\n" ); document.write( "start with:
\n" ); document.write( "sqrt(x+3) <= (x+1)
\n" ); document.write( "square both sides to get:
\n" ); document.write( "(x+3) <= (x+1)^2
\n" ); document.write( "simplify to get:
\n" ); document.write( "(x+3) <= x^2 + 2x + 1
\n" ); document.write( "subtract (x+3) from both side to get:
\n" ); document.write( "0 <= x^2 + 2x + 1 - (x+3)
\n" ); document.write( "simplify to get:
\n" ); document.write( "0 <= x^2 + 2x + 1 - x - 3
\n" ); document.write( "simplify further to get:
\n" ); document.write( "0 <= x^2 + x - 2
\n" ); document.write( "factor the equality portion of this equation to get:
\n" ); document.write( "0 = (x+2) * (x-1)
\n" ); document.write( "solve for x to get:
\n" ); document.write( "x = -2 or x = 1
\n" ); document.write( "replace x in the original equation to confirm the solution is good.
\n" ); document.write( "when x = 1:
\n" ); document.write( "sqrt(x+3) <= x+1 becomes sqrt(4) <= 2 which becomes 2 <= 2 which is true.
\n" ); document.write( "when x = -2:
\n" ); document.write( "sqrt(x+3) <= x+1 becomes sqrt(1) <= -2+1 which becomes 1 <= -1 which is false.
\n" ); document.write( "only x = 1 is a good solution.
\n" ); document.write( "x = -2 is an extraneous solution that cannot be confirmed in the original equation and is therefore not a solution.
\n" ); document.write( "the inequality is true when x is greater than or equal to 1.
\n" ); document.write( "confirm this by checking your inequality in the regions of -3 <= x <= 1 and x > 1.
\n" ); document.write( "since you graphed this, you don't need to, but you can confirm anyway just to make sure you didn't mess anything up.
\n" ); document.write( "when x = -3, the inequality becomes 0 <= -2 which is false.
\n" ); document.write( "when x = -2, the inequality becomes 1 <= -1 which is false.
\n" ); document.write( "when x = -1, the inequality becomes sqrt(2) <= 0 which is false.
\n" ); document.write( "when x = 0, the inequality becomes sqrt(3) <= 1 which is false.
\n" ); document.write( "wen x = 1, the inequality becomes sqrt(4) <= 2 which is true.
\n" ); document.write( "when x = 2, the inequality becomes sqrt(5) <= 3 which is true.
\n" ); document.write( "when x = 3, the inequality becomes sqrt(6) <= 4 which is true.
\n" ); document.write( "the calculations confirm the graph.
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