document.write( "Question 67976: Could someone please please help me with this:\r
\n" ); document.write( "\n" ); document.write( "f(x)= (x²-1)(x-1). \r
\n" ); document.write( "\n" ); document.write( "1.Show why f(x)is greater or equal to 0 in each of the following cases:
\n" ); document.write( "a)x>1
\n" ); document.write( "b)0\n" ); document.write( "c)-1\n" ); document.write( "2.Deduce that x^3+1>x^2+x for all x>-1.\r
\n" ); document.write( "\n" ); document.write( "Thank you very for your great help!!!!!!! \n" ); document.write( "
Algebra.Com's Answer #48338 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Could someone please please help me with this:\r\n" );
document.write( "\r\n" );
document.write( "f(x)= (x²-1)(x-1). \r\n" );
document.write( "\r\n" );
document.write( "First completely factor the expression for f(x)\r\n" );
document.write( "\r\n" );
document.write( "f(x) = (x-1)(x+1)(x-1)\r\n" );
document.write( "\r\n" );
document.write( "f(x) = (x-1)²(x+1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1.Show why f(x)is greater or equal to 0 in each of the following cases:\r\n" );
document.write( "\r\n" );
document.write( "a)  x > 1\r\n" );
document.write( "\r\n" );
document.write( "1.     x > 1                    Given\r\n" );
document.write( "\r\n" );
document.write( "2.   x+1 > 2                    Added 1 to both sides of 1\r\n" );
document.write( "\r\n" );
document.write( "3.  (x-1)²(x+1) > 2(x-1)²       Multiplied both sides of 2 by\r\n" );
document.write( "                                nonnegative quantity (x-1)², which is\r\n" );
document.write( "                                nonnegative because the square of any\r\n" );
document.write( "                                quantity is nonnegative.\r\n" );
document.write( "\r\n" );
document.write( "4.  f(x) > 2(x-1)²              Substituted f(x) for (x-1)²(x+1) in 3.\r\n" );
document.write( "\r\n" );
document.write( "5.  f(x) > 0                    Because 4 tells us f(x) is greater than or\r\n" );
document.write( "                                equal to twice the square of a quantity, which\r\n" );
document.write( "                                is nonnegative.\r\n" );
document.write( "\r\n" );
document.write( "----------------------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "b)   0 < x < 1\r\n" );
document.write( "\r\n" );
document.write( "1.   0 < x < 1                          Given     \r\n" );
document.write( "\r\n" );
document.write( "2.  0+1 < x+1 < 1+1                     Added 1 to all 3 sides of 1\r\n" );
document.write( "\r\n" );
document.write( "3.    1 < x+1 < 2                       Simplified 2\r\n" );
document.write( "\r\n" );
document.write( "4.  (x-1)²·1 < (x-1)²(x+1) < (x-1)²·2   Multiplied all 3 sides of 3\r\n" );
document.write( "                                        by non-negative quantity (x-1)², \r\n" );
document.write( "                                        which is known to be non-negative\r\n" );
document.write( "                                        because any quantity squared is\r\n" );
document.write( "                                        nonnegative.\r\n" );
document.write( "\r\n" );
document.write( "5. (x-1)² < (x-1)²(x-1) < 2(x+1)²       Simplified 4\r\n" );
document.write( "\r\n" );
document.write( "6. (x-1)²(x+1) > (x-1)²                 Writing 5 equivalently. (Don't need\r\n" );
document.write( "                                        right side of 5.)  \r\n" );
document.write( "\r\n" );
document.write( "7.  f(x) > (x-1)²                       Substituting f(x) for (x-1)²(x+1)\r\n" );
document.write( "\r\n" );
document.write( "8.  f(x) > 0                            Because 7 tells us f(x) is greater\r\n" );
document.write( "                                        than a square quantity, and a \r\n" );
document.write( "                                        squared quantity is greater than or \r\n" );
document.write( "                                        equal to 0. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "--------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "c) -1 < x < 0\r\n" );
document.write( "\r\n" );
document.write( "1.  -1 < x < 0                          Given     \r\n" );
document.write( "\r\n" );
document.write( "2. -1+1 < x+1 < 0+1                     Added 1 to all 3 sides of 1\r\n" );
document.write( "\r\n" );
document.write( "3.    0 < x+1 < 1                       Simplified 2\r\n" );
document.write( "\r\n" );
document.write( "4.  (x-1)²·0 > (x-1)²(x+1) > (x-1)²·1   Multiplied all 3 sides of 3\r\n" );
document.write( "                                        by non-negative quantity (x-1)², \r\n" );
document.write( "                                        which is known to be non-negative\r\n" );
document.write( "                                        because any quantity squared is\r\n" );
document.write( "                                        nonnegative.\r\n" );
document.write( "\r\n" );
document.write( "5.   0 > (x-1)²(x+1) > (x-1)²           Simplified 4\r\n" );
document.write( "\r\n" );
document.write( "6.  (x-1)²(x+1) > 0                     Writing 5 equivalently. (Don't need\r\n" );
document.write( "                                        right side of 5.)  \r\n" );
document.write( "\r\n" );
document.write( "7.  f(x) > 0                            Substituting f(x) for (x-1)²(x+1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "------\r\n" );
document.write( "2.Deduce that\r\n" );
document.write( "\r\n" );
document.write( " x³+1 > x²+x for all x > -1\r\n" );
document.write( "\r\n" );
document.write( "This isn't true because when x = 1, the inequality \r\n" );
document.write( "becomes 2 > 2 which is false.\r\n" );
document.write( "\r\n" );
document.write( "You must have meant \r\n" );
document.write( "\r\n" );
document.write( " x³+1 > x²+x for all x > -1\r\n" );
document.write( "\r\n" );
document.write( "Factor both sides of the inequality, to see what\r\n" );
document.write( "we need to prove.  We need to prove this:\r\n" );
document.write( "\r\n" );
document.write( " (x+1)(x²-x+1) > x(x+1)\r\n" );
document.write( "\r\n" );
document.write( "1. x > -1                    Given\r\n" );
document.write( "\r\n" );
document.write( "2. x+1 > 0                   Added 1 to both sides of 1\r\n" );
document.write( " \r\n" );
document.write( "Now we need to work backwards and look at what we need.\r\n" );
document.write( "We need to show that x²-x+1 > x so we can multiply both\r\n" );
document.write( "sides of 2  by x+1 which we know is positive from 2.  But\r\n" );
document.write( "that inequality is equivalent to x²-2x+1 > 0 by adding\r\n" );
document.write( "-x to both sides.  Then that is equivalent to (x-1)² > 0.\r\n" );
document.write( "But that is true because any squared quantity is \r\n" );
document.write( "nonnegative. So now we reverse our thinking here:\r\n" );
document.write( "\r\n" );
document.write( "3. (x-1)² > 0               The square of any quantity is\r\n" );
document.write( "                            nonnegative.\r\n" );
document.write( "\r\n" );
document.write( "4. x²-2x+1 > 0              Squared x-1 in 3. \r\n" );
document.write( "\r\n" );
document.write( "5. x²-x+1 > x               Added x to both sides.  \r\n" );
document.write( "\r\n" );
document.write( "6. (x+1)(x²-x+1) > x(x+1)   Multiplied both sides of 5 by positive\r\n" );
document.write( "                            quantity (x+1), which we know is \r\n" );
document.write( "                            positive since 2 tells us it is greater\r\n" );
document.write( "                            than 0.\r\n" );
document.write( "\r\n" );
document.write( "That's what we had to prove. \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );