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Answer :
To solve the inequality [tex]\(x^3 + 5x^2 - 9x < 45\)[/tex], we start by simplifying it. Bring all terms to one side of the inequality:
[tex]\[x^3 + 5x^2 - 9x - 45 < 0.\][/tex]
Now, we'll find the intervals where the expression is less than zero by analyzing the critical points where the expression equals zero. To do this, we set up the equation:
[tex]\[ x^3 + 5x^2 - 9x - 45 = 0. \][/tex]
Solving this cubic equation will give us the roots. These roots will help us identify the critical points which we use to divide the number line into intervals. Within each interval, the sign of the expression [tex]\(x^3 + 5x^2 - 9x - 45\)[/tex] will be either positive or negative.
For this equation, the roots turn out to be:
1. [tex]\( x = -5 \)[/tex]
2. [tex]\( x = -3 \)[/tex]
3. [tex]\( x = 3 \)[/tex]
These roots divide the number line into four intervals:
1. [tex]\( (-\infty, -5) \)[/tex]
2. [tex]\( (-5, -3) \)[/tex]
3. [tex]\( (-3, 3) \)[/tex]
4. [tex]\( (3, \infty) \)[/tex]
Next, we test a number from each interval to determine whether the expression is positive or negative in that interval.
- In interval [tex]\( (-\infty, -5) \)[/tex], choose [tex]\( x = -6 \)[/tex].
- In interval [tex]\( (-5, -3) \)[/tex], choose [tex]\( x = -4 \)[/tex].
- In interval [tex]\( (-3, 3) \)[/tex], choose [tex]\( x = 0 \)[/tex].
- In interval [tex]\( (3, \infty) \)[/tex], choose [tex]\( x = 4 \)[/tex].
Evaluating the expression at these test points will show us where the expression is negative:
1. [tex]\( (-\infty, -5) \)[/tex]: The expression is negative.
2. [tex]\( (-5, -3) \)[/tex]: The expression is positive.
3. [tex]\( (-3, 3) \)[/tex]: The expression is negative.
4. [tex]\( (3, \infty) \)[/tex]: The expression is positive.
Therefore, the solution to the inequality [tex]\(x^3 + 5x^2 - 9x < 45\)[/tex] is the set of [tex]\(x\)[/tex]-values where the expression is negative, which occurs in the intervals:
[tex]\[ (-\infty, -5) \cup (-3, 3). \][/tex]
This means that for the values of [tex]\(x\)[/tex] in these intervals, the original inequality holds true.
[tex]\[x^3 + 5x^2 - 9x - 45 < 0.\][/tex]
Now, we'll find the intervals where the expression is less than zero by analyzing the critical points where the expression equals zero. To do this, we set up the equation:
[tex]\[ x^3 + 5x^2 - 9x - 45 = 0. \][/tex]
Solving this cubic equation will give us the roots. These roots will help us identify the critical points which we use to divide the number line into intervals. Within each interval, the sign of the expression [tex]\(x^3 + 5x^2 - 9x - 45\)[/tex] will be either positive or negative.
For this equation, the roots turn out to be:
1. [tex]\( x = -5 \)[/tex]
2. [tex]\( x = -3 \)[/tex]
3. [tex]\( x = 3 \)[/tex]
These roots divide the number line into four intervals:
1. [tex]\( (-\infty, -5) \)[/tex]
2. [tex]\( (-5, -3) \)[/tex]
3. [tex]\( (-3, 3) \)[/tex]
4. [tex]\( (3, \infty) \)[/tex]
Next, we test a number from each interval to determine whether the expression is positive or negative in that interval.
- In interval [tex]\( (-\infty, -5) \)[/tex], choose [tex]\( x = -6 \)[/tex].
- In interval [tex]\( (-5, -3) \)[/tex], choose [tex]\( x = -4 \)[/tex].
- In interval [tex]\( (-3, 3) \)[/tex], choose [tex]\( x = 0 \)[/tex].
- In interval [tex]\( (3, \infty) \)[/tex], choose [tex]\( x = 4 \)[/tex].
Evaluating the expression at these test points will show us where the expression is negative:
1. [tex]\( (-\infty, -5) \)[/tex]: The expression is negative.
2. [tex]\( (-5, -3) \)[/tex]: The expression is positive.
3. [tex]\( (-3, 3) \)[/tex]: The expression is negative.
4. [tex]\( (3, \infty) \)[/tex]: The expression is positive.
Therefore, the solution to the inequality [tex]\(x^3 + 5x^2 - 9x < 45\)[/tex] is the set of [tex]\(x\)[/tex]-values where the expression is negative, which occurs in the intervals:
[tex]\[ (-\infty, -5) \cup (-3, 3). \][/tex]
This means that for the values of [tex]\(x\)[/tex] in these intervals, the original inequality holds true.
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