solve the inequality: $w^{2}<11w - 18$\ngive your answer in interval notation. enter dne if there is no…

solve the inequality: $w^{2}<11w - 18$\ngive your answer in interval notation. enter dne if there is no solution.

solve the inequality: $w^{2}<11w - 18$\ngive your answer in interval notation. enter dne if there is no solution.

Answer

Explanation:

Step1: Rearrange the inequality

Move all terms to one side: (w^{2}-11w + 18<0).

Step2: Factor the quadratic expression

Factor (w^{2}-11w + 18). We need two numbers that multiply to (18) and add up to (-11). These numbers are (-2) and (-9). So, (w^{2}-11w + 18=(w - 2)(w - 9)<0).

Step3: Find the critical points

Set ((w - 2)(w - 9)=0). The critical points are (w = 2) and (w = 9).

Step4: Test intervals

We have three intervals to test: ((-\infty,2)), ((2,9)), and ((9,\infty)).

  • For (w=0) (in ((-\infty,2))): ((0 - 2)(0 - 9)=18>0).
  • For (w = 5) (in ((2,9))): ((5 - 2)(5 - 9)=-12<0).
  • For (w = 10) (in ((9,\infty))): ((10 - 2)(10 - 9)=8>0).

Answer:

((2,9))