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Answer :
To solve the inequality [tex]\(3x^3 > 27x^2\)[/tex], we can follow these steps:
1. Simplify the inequality:
Start by factoring out the common term on both sides. Notice that both terms on the left and right side of the inequality have a factor of [tex]\(3x^2\)[/tex].
[tex]\[
3x^3 > 27x^2
\][/tex]
Divide both sides by [tex]\(3x^2\)[/tex] (assuming [tex]\(x \neq 0\)[/tex] to avoid division by zero):
[tex]\[
x > 9
\][/tex]
2. Consider the values of x:
The solution [tex]\(x > 9\)[/tex] implies that the value of [tex]\(x\)[/tex] must be greater than 9. However, we initially assumed [tex]\(x \neq 0\)[/tex] for the division to be valid. Let's consider this:
- If [tex]\(x = 0\)[/tex], the original inequality becomes [tex]\(0 > 0\)[/tex], which is not true.
- If [tex]\(x < 0\)[/tex], then [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] will have different signs because [tex]\(x^3\)[/tex] is negative and [tex]\(x^2\)[/tex] is positive. Therefore, [tex]\(3x^3\)[/tex] will always be less than [tex]\(27x^2\)[/tex], making the inequality false for negative [tex]\(x\)[/tex].
3. Conclude the solution range:
Since values where [tex]\(x \leq 0\)[/tex] do not satisfy the inequality and values where [tex]\(x > 9\)[/tex] do satisfy it, the complete solution to the inequality [tex]\(3x^3 > 27x^2\)[/tex] is:
[tex]\[
x > 9
\][/tex]
Thus, the solution set of the inequality is [tex]\(x > 9\)[/tex], which means all values of [tex]\(x\)[/tex] greater than 9 satisfy the given inequality.
1. Simplify the inequality:
Start by factoring out the common term on both sides. Notice that both terms on the left and right side of the inequality have a factor of [tex]\(3x^2\)[/tex].
[tex]\[
3x^3 > 27x^2
\][/tex]
Divide both sides by [tex]\(3x^2\)[/tex] (assuming [tex]\(x \neq 0\)[/tex] to avoid division by zero):
[tex]\[
x > 9
\][/tex]
2. Consider the values of x:
The solution [tex]\(x > 9\)[/tex] implies that the value of [tex]\(x\)[/tex] must be greater than 9. However, we initially assumed [tex]\(x \neq 0\)[/tex] for the division to be valid. Let's consider this:
- If [tex]\(x = 0\)[/tex], the original inequality becomes [tex]\(0 > 0\)[/tex], which is not true.
- If [tex]\(x < 0\)[/tex], then [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex] will have different signs because [tex]\(x^3\)[/tex] is negative and [tex]\(x^2\)[/tex] is positive. Therefore, [tex]\(3x^3\)[/tex] will always be less than [tex]\(27x^2\)[/tex], making the inequality false for negative [tex]\(x\)[/tex].
3. Conclude the solution range:
Since values where [tex]\(x \leq 0\)[/tex] do not satisfy the inequality and values where [tex]\(x > 9\)[/tex] do satisfy it, the complete solution to the inequality [tex]\(3x^3 > 27x^2\)[/tex] is:
[tex]\[
x > 9
\][/tex]
Thus, the solution set of the inequality is [tex]\(x > 9\)[/tex], which means all values of [tex]\(x\)[/tex] greater than 9 satisfy the given inequality.
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