Answer
(a) 1. move all items to one side; 2. factorize both the numerator and the denominator;
3. list all cut points; 4. build a sign table; 5. check endpoints and make conclusions based on the table.
(b) Cut points separate regions of the function where the signs may change. To solve a rational inequality, use cut points to build up a sign table where the signs of the function are evaluated, see answer (a) for details.
(c) $[-1,9]$
Work Step by Step
(a) Explain how to solve a polynomial inequality.
1. move all items to one side; 2. factorize both the numerator and the denominator;
3. list all cut points; 4. build a sign table; 5. check endpoints and make conclusions based on the table.
(b) What are the cut points of a rational function? Explain how to solve a rational inequality.
Cut points separate regions of the function where the signs may change. To solve a rational inequality, use cut points to build up a sign table where the signs of the function are evaluated, see answer (a) for details.
(c) $f(x)=x^2-8x-9=(x-9)(x+1)\leq0$, cut points $-1,9$, make a table as shown and we
conclude the solution as $[-1,9]$