Question
Solve the quadratic inequality −x2−9x−14≥0 by factoring.(1 point)%0D%0AResponses%0D%0A%0D%0A(−∞,−7]∪[−2,∞)%0D%0Aleft parenthesis negative infinity comma negative 7 right bracket union left bracket negative 2 comma infinity right parenthesis%0D%0A%0D%0A[−7,−2]%0D%0Aleft bracket negative 7 comma negative 2 right bracket%0D%0A%0D%0A(−∞,−2]∪[7,∞)%0D%0Aleft parenthesis negative infinity comma negative 2 right bracket union left bracket 7 comma infinity right parenthesis%0D%0A%0D%0A[−7,2]
Answers
Answered by
GPT-4o mini
To solve the quadratic inequality \(-x^2 - 9x - 14 \geq 0\) by factoring, we will first rewrite the inequality for easier analysis.
Start by multiplying the entire inequality by -1 (remember to reverse the inequality sign):
\[
x^2 + 9x + 14 \leq 0
\]
Next, we need to factor the quadratic \(x^2 + 9x + 14\). We need two numbers that multiply to 14 and add to 9. The numbers 7 and 2 satisfy this requirement, so we can factor it as:
\[
(x + 7)(x + 2) \leq 0
\]
Next, we find the critical points by setting each factor equal to zero:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
Now we will test intervals around these critical points: \((-∞, -7)\), \((-7, -2)\), and \((-2, ∞)\).
1. **Interval \((-∞, -7)\)**: Choose \(x = -8\):
\[
(-8 + 7)(-8 + 2) = (-1)(-6) = 6 \quad (> 0)
\]
2. **Interval \((-7, -2)\)**: Choose \(x = -5\):
\[
(-5 + 7)(-5 + 2) = (2)(-3) = -6 \quad (< 0)
\]
3. **Interval \((-2, ∞)\)**: Choose \(x = 0\):
\[
(0 + 7)(0 + 2) = (7)(2) = 14 \quad (> 0)
\]
The inequality is satisfied (i.e., the product is less than or equal to zero) on the interval \((-7, -2)\).
Since we also need to include the points where the product is zero, we have:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
Thus, the solution to the inequality \(x^2 + 9x + 14 \leq 0\) is:
\[
[-7, -2]
\]
Hence, the final answer is:
\[
[-7, -2]
\]
Start by multiplying the entire inequality by -1 (remember to reverse the inequality sign):
\[
x^2 + 9x + 14 \leq 0
\]
Next, we need to factor the quadratic \(x^2 + 9x + 14\). We need two numbers that multiply to 14 and add to 9. The numbers 7 and 2 satisfy this requirement, so we can factor it as:
\[
(x + 7)(x + 2) \leq 0
\]
Next, we find the critical points by setting each factor equal to zero:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
Now we will test intervals around these critical points: \((-∞, -7)\), \((-7, -2)\), and \((-2, ∞)\).
1. **Interval \((-∞, -7)\)**: Choose \(x = -8\):
\[
(-8 + 7)(-8 + 2) = (-1)(-6) = 6 \quad (> 0)
\]
2. **Interval \((-7, -2)\)**: Choose \(x = -5\):
\[
(-5 + 7)(-5 + 2) = (2)(-3) = -6 \quad (< 0)
\]
3. **Interval \((-2, ∞)\)**: Choose \(x = 0\):
\[
(0 + 7)(0 + 2) = (7)(2) = 14 \quad (> 0)
\]
The inequality is satisfied (i.e., the product is less than or equal to zero) on the interval \((-7, -2)\).
Since we also need to include the points where the product is zero, we have:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
Thus, the solution to the inequality \(x^2 + 9x + 14 \leq 0\) is:
\[
[-7, -2]
\]
Hence, the final answer is:
\[
[-7, -2]
\]
Answer
Just figure it out yourself :/
Answered by
GPT-4o mini
Sure! Let's go through the solution step by step for the inequality \(-x^2 - 9x - 14 \geq 0\).
1. **Rewrite the inequality**: To make handling easier, multiply through by -1 (reversing the inequality):
\[
x^2 + 9x + 14 \leq 0
\]
2. **Factor the quadratic**: We need to factor \(x^2 + 9x + 14\). We look for two numbers that multiply to 14 and add to 9, which are 7 and 2. Thus, we can write:
\[
(x + 7)(x + 2) \leq 0
\]
3. **Identify the critical points**: Set each factor to zero to find the critical points:
\[
x + 7 = 0 \implies x = -7
\]
\[
x + 2 = 0 \implies x = -2
\]
4. **Test the intervals**: The critical points divide the number line into the intervals: \((-∞, -7)\), \((-7, -2)\), and \((-2, ∞)\). We need to test each interval:
- For the interval \((-∞, -7)\), say \(x = -8\):
\[
(-8 + 7)(-8 + 2) = (-1)(-6) = 6 \quad (> 0)
\]
- For the interval \((-7, -2)\), say \(x = -5\):
\[
(-5 + 7)(-5 + 2) = (2)(-3) = -6 \quad (< 0)
\]
- For the interval \((-2, ∞)\), say \(x = 0\):
\[
(0 + 7)(0 + 2) = (7)(2) = 14 \quad (> 0)
\]
5. **Solution**: The product \((x + 7)(x + 2) \leq 0\) is satisfied in the interval \([-7, -2]\) where it is either zero or negative.
Final answer:
\[
[-7, -2]
\]
If you have any more questions or need further clarification, feel free to ask!
1. **Rewrite the inequality**: To make handling easier, multiply through by -1 (reversing the inequality):
\[
x^2 + 9x + 14 \leq 0
\]
2. **Factor the quadratic**: We need to factor \(x^2 + 9x + 14\). We look for two numbers that multiply to 14 and add to 9, which are 7 and 2. Thus, we can write:
\[
(x + 7)(x + 2) \leq 0
\]
3. **Identify the critical points**: Set each factor to zero to find the critical points:
\[
x + 7 = 0 \implies x = -7
\]
\[
x + 2 = 0 \implies x = -2
\]
4. **Test the intervals**: The critical points divide the number line into the intervals: \((-∞, -7)\), \((-7, -2)\), and \((-2, ∞)\). We need to test each interval:
- For the interval \((-∞, -7)\), say \(x = -8\):
\[
(-8 + 7)(-8 + 2) = (-1)(-6) = 6 \quad (> 0)
\]
- For the interval \((-7, -2)\), say \(x = -5\):
\[
(-5 + 7)(-5 + 2) = (2)(-3) = -6 \quad (< 0)
\]
- For the interval \((-2, ∞)\), say \(x = 0\):
\[
(0 + 7)(0 + 2) = (7)(2) = 14 \quad (> 0)
\]
5. **Solution**: The product \((x + 7)(x + 2) \leq 0\) is satisfied in the interval \([-7, -2]\) where it is either zero or negative.
Final answer:
\[
[-7, -2]
\]
If you have any more questions or need further clarification, feel free to ask!