Question

practice questions: problem 9 (1 point) results for this submission entered answer preview (-infinity,-9 ∪ 9,infinity) (-∞, -9 ∪ 9, ∞) the answer above is not correct. solve the following inequality. express the answer in interval notation. ( x^4 > 81x^2 ) answer: ( (-infty, -9 cup 9, infty) ) your score was recorded. you have attempted this problem 1 time. you have 2 attempts left before new version will be requested. you received a score of 0% for this attempt. your overall recorded score is 0%. you have unlimited attempts remaining.

Explanation:

Step1: Rearrange the inequality

Subtract \(81x^2\) from both sides to get \(x^4 - 81x^2>0\).

Step2: Factor the left - hand side

Factor out \(x^2\) first: \(x^2(x^2 - 81)>0\). Then, since \(x^2 - 81=(x - 9)(x + 9)\) (by the difference of squares formula \(a^2-b^2=(a - b)(a + b)\) where \(a = x\) and \(b = 9\)), the inequality becomes \(x^2(x - 9)(x + 9)>0\).

Step3: Analyze the critical points

The critical points are found by setting each factor equal to zero: \(x^2 = 0\) gives \(x = 0\), \(x-9=0\) gives \(x = 9\), and \(x + 9=0\) gives \(x=-9\). These critical points divide the real number line into four intervals: \((-\infty,-9)\), \((-9,0)\), \((0,9)\), and \((9,\infty)\).

Step4: Test intervals

• For the interval \((-\infty,-9)\), let's choose a test point, say \(x=-10\). Then \(x^2=(-10)^2 = 100>0\), \(x - 9=-10 - 9=-19<0\), \(x + 9=-10 + 9=-1<0\). The product \(x^2(x - 9)(x + 9)=(100)\times(-19)\times(-1)=1900>0\), so the inequality holds on \((-\infty,-9)\).
• For the interval \((-9,0)\), let's choose a test point, say \(x=-1\). Then \(x^2=(-1)^2 = 1>0\), \(x - 9=-1-9=-10<0\), \(x + 9=-1 + 9 = 8>0\). The product \(x^2(x - 9)(x + 9)=(1)\times(-10)\times(8)=-80<0\), so the inequality does not hold on \((-9,0)\).
• For the interval \((0,9)\), let's choose a test point, say \(x = 1\). Then \(x^2=1^2 = 1>0\), \(x - 9=1-9=-8<0\), \(x + 9=1 + 9 = 10>0\). The product \(x^2(x - 9)(x + 9)=(1)\times(-8)\times(10)=-80<0\), so the inequality does not hold on \((0,9)\).
• For the interval \((9,\infty)\), let's choose a test point, say \(x = 10\). Then \(x^2=10^2 = 100>0\), \(x - 9=10 - 9 = 1>0\), \(x + 9=10 + 9 = 19>0\). The product \(x^2(x - 9)(x + 9)=(100)\times(1)\times(19)=1900>0\), so the inequality holds on \((9,\infty)\).

We also note that when \(x = 0\), \(x^4=0\) and \(81x^2 = 0\), so \(x^4=81x^2\) and the inequality \(x^4>81x^2\) is not satisfied. When \(x=\pm9\), \(x^4=( \pm 9)^4=6561\) and \(81x^2=81\times81 = 6561\), so \(x^4 = 81x^2\) and the inequality is not satisfied.

Answer:

\((-\infty,-9)\cup(9,\infty)\)