document.write( "Question 629494: What are the values of x for which 8-x^2 < 5?\r
\n" ); document.write( "\n" ); document.write( "I tried solving for x and got x < sqrt(3). That is only one piece in this puzzle, though. How would I go about answering this question?\r
\n" ); document.write( "\n" ); document.write( "This problem comes from a practice test I took today in preparation for an upcoming math placement exam.\r
\n" ); document.write( "\n" ); document.write( "Thank you!\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #396316 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Actually, your \"one piece\" is not right. I suspect you multiplied or divided both sides by a negative at some point and forgot the special rule that tells you to reverse the inequality.

\n" ); document.write( "There are several ways to figure this out. One of them would be...
\n" ); document.write( "Make one side of the inequality zero. The fast way to do this would be to subtract 5 from each side. But I prefer to avoid the special rule. So I like to make sure the variable term has a positive coefficient. So I am going to subtract 8 and add \"x%5E2\" on both sides:
\n" ); document.write( "\"0+%3C+x%5E2-3\"
\n" ); document.write( "Now I'm going to use the difference of squares pattern, \"a%5E2-b%5E2=%28a%2Bb%29%28a-b%29\" to factor the right side, You might wonder: \"But 3 is not a perfect square!?\" And you'd be right. It's not the square of a whole number. But every positive number is the square of something. In this case 3 is the square of \"sqrt%283%29\". So we can factor the right side with this pattern:
\n" ); document.write( "\"0+%3C+%28x+%2B+sqrt%283%29%29%28x-sqrt%283%29%29\"

\n" ); document.write( "The reasons we got a zero and then factored are that the inequality above tells us something important. It tells us that a product (the right side) is greater than zero. A synonym for \"greater than zero\" is positive. So this tells us that a product of two numbers is positive. You have probably known for a long time that to get a positive result when multiplying two numbers you must have:
  • Two positive numbers; or
  • Two negative numbers
So our solution to this problem will be \"all the x values that make both factors positive or both factors negative.\" All we have to do is express this logic in the form a inequalities we can solve.

\n" ); document.write( "For \"both factors are positive\" I hope the following makes sense:
\n" ); document.write( "(x+sqrt(3) > 0 and x-sqrt(3)>0)
\n" ); document.write( "For \"both factors are negative\" I hope the following makes sense:
\n" ); document.write( "(x+sqrt(3) < 0 and x-sqrt(3)<0)
\n" ); document.write( "For \"both factors positive or both factors negative\" I hope that the following makes sense:
\n" ); document.write( "(x+sqrt(3) > 0 and x-sqrt(3)>0) or (x+sqrt(3) < 0 and x-sqrt(3)<0)
\n" ); document.write( "Now we just solve these:
\n" ); document.write( "(x > -sqrt(3) and x>sqrt(3)) or (x < -sqrt(3) and x
\n" ); document.write( "The first pair both say that x is greater than something. With the \"and\" between them we are interested only in x's that are greater than both. I hope it makes sense that if x is greater than the larger number then it would automatically be greater than the smaller number, too. So \"x+%3E+sqrt%283%29\" is all we need.

\n" ); document.write( "The second pair both say that x is less than something. With the \"and\" between them we are interested only in x's that are less than both. I hope it makes sense that if x is less than the smaller number then it would automatically be less than the larger number, too. So \"x+%3C+-sqrt%283%29\" is all we need.

\n" ); document.write( "So our simplified solution is:
\n" ); document.write( "(x>sqrt(3)) or (x < -sqrt(3)) \n" ); document.write( "
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