Answer :
Sure, let's solve the inequality step by step:
We start with the inequality:
[tex]\[ x^3 + 3x^2 - 16x \leq 48 \][/tex]
1. Rewrite the inequality to bring all terms to one side:
[tex]\[ x^3 + 3x^2 - 16x - 48 \leq 0 \][/tex]
2. Factor the polynomial [tex]\( x^3 + 3x^2 - 16x - 48 \)[/tex]:
The given polynomial is cubic, and factoring cubic polynomials involves finding the roots. According to the information, the roots of the equation are:
[tex]\[ x = -4, x = -3, x = 4 \][/tex]
Therefore, we can express the polynomial as:
[tex]\[ (x + 4)(x + 3)(x - 4) \leq 0 \][/tex]
This was obtained by using the roots and rewriting them as factors.
3. Determine the intervals by setting each factor to zero:
The roots divide the number line into four intervals:
- [tex]\( (-\infty, -4] \)[/tex]
- [tex]\( (-4, -3] \)[/tex]
- [tex]\( (-3, 4) \)[/tex]
- [tex]\( (4, \infty) \)[/tex]
4. Test each interval to check where the inequality holds true. You can choose a test point from each interval:
- For [tex]\( (-\infty, -4) \)[/tex], pick [tex]\( x = -5 \)[/tex]:
[tex]\[ (-5 + 4)(-5 + 3)(-5 - 4) = -1 \cdot -2 \cdot -9 = -18 \leq 0 \][/tex]
Therefore, this interval satisfies the inequality.
- For [tex]\( (-4, -3) \)[/tex], pick [tex]\( x = -3.5 \)[/tex]:
[tex]\[ (-3.5 + 4)(-3.5 + 3)(-3.5 - 4) = 0.5 \cdot -0.5 \cdot -7.5 = 1.875 \not\leq 0 \][/tex]
Therefore, this interval does not satisfy the inequality.
- For [tex]\( (-3, 4) \)[/tex], pick [tex]\( x = 0 \)[/tex]:
[tex]\[ (0 + 4)(0 + 3)(0 - 4) = 4 \cdot 3 \cdot -4 = -48 \leq 0 \][/tex]
Therefore, this interval satisfies the inequality.
- For [tex]\( (4, \infty) \)[/tex], pick [tex]\( x = 5 \)[/tex]:
[tex]\[ (5 + 4)(5 + 3)(5 - 4) = 9 \cdot 8 \cdot 1 = 72 \not\leq 0 \][/tex]
Therefore, this interval does not satisfy the inequality.
5. Summarize the solution, including the points where the expression equals zero:
- The critical points are [tex]\( x = -4, -3, \)[/tex] and [tex]\( 4 \)[/tex]. At these points, the expression equals zero which satisfies the inequality [tex]\( \leq 0 \)[/tex].
So, the solution to the inequality [tex]\( x^3 + 3x^2 - 16x \leq 48 \)[/tex] is:
[tex]\[ x \in (-\infty, -4] \cup [-3, 4] \][/tex]
We start with the inequality:
[tex]\[ x^3 + 3x^2 - 16x \leq 48 \][/tex]
1. Rewrite the inequality to bring all terms to one side:
[tex]\[ x^3 + 3x^2 - 16x - 48 \leq 0 \][/tex]
2. Factor the polynomial [tex]\( x^3 + 3x^2 - 16x - 48 \)[/tex]:
The given polynomial is cubic, and factoring cubic polynomials involves finding the roots. According to the information, the roots of the equation are:
[tex]\[ x = -4, x = -3, x = 4 \][/tex]
Therefore, we can express the polynomial as:
[tex]\[ (x + 4)(x + 3)(x - 4) \leq 0 \][/tex]
This was obtained by using the roots and rewriting them as factors.
3. Determine the intervals by setting each factor to zero:
The roots divide the number line into four intervals:
- [tex]\( (-\infty, -4] \)[/tex]
- [tex]\( (-4, -3] \)[/tex]
- [tex]\( (-3, 4) \)[/tex]
- [tex]\( (4, \infty) \)[/tex]
4. Test each interval to check where the inequality holds true. You can choose a test point from each interval:
- For [tex]\( (-\infty, -4) \)[/tex], pick [tex]\( x = -5 \)[/tex]:
[tex]\[ (-5 + 4)(-5 + 3)(-5 - 4) = -1 \cdot -2 \cdot -9 = -18 \leq 0 \][/tex]
Therefore, this interval satisfies the inequality.
- For [tex]\( (-4, -3) \)[/tex], pick [tex]\( x = -3.5 \)[/tex]:
[tex]\[ (-3.5 + 4)(-3.5 + 3)(-3.5 - 4) = 0.5 \cdot -0.5 \cdot -7.5 = 1.875 \not\leq 0 \][/tex]
Therefore, this interval does not satisfy the inequality.
- For [tex]\( (-3, 4) \)[/tex], pick [tex]\( x = 0 \)[/tex]:
[tex]\[ (0 + 4)(0 + 3)(0 - 4) = 4 \cdot 3 \cdot -4 = -48 \leq 0 \][/tex]
Therefore, this interval satisfies the inequality.
- For [tex]\( (4, \infty) \)[/tex], pick [tex]\( x = 5 \)[/tex]:
[tex]\[ (5 + 4)(5 + 3)(5 - 4) = 9 \cdot 8 \cdot 1 = 72 \not\leq 0 \][/tex]
Therefore, this interval does not satisfy the inequality.
5. Summarize the solution, including the points where the expression equals zero:
- The critical points are [tex]\( x = -4, -3, \)[/tex] and [tex]\( 4 \)[/tex]. At these points, the expression equals zero which satisfies the inequality [tex]\( \leq 0 \)[/tex].
So, the solution to the inequality [tex]\( x^3 + 3x^2 - 16x \leq 48 \)[/tex] is:
[tex]\[ x \in (-\infty, -4] \cup [-3, 4] \][/tex]
