Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the values of [tex]\( x \)[/tex] that satisfy both inequalities:
[tex]\[ \begin{array}{l} x^2-2 x-24<0 \\ -x^2+9 x-14 \geq 0 \end{array} \][/tex]
we will solve each inequality separately and then find their intersection.
### Step 1: Solve the First Inequality
The first inequality is:
[tex]\[ x^2 - 2x - 24 < 0 \][/tex]
1. Find the roots of the equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex]:
[tex]\[ x^2 - 2x - 24 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 96}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{100}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm 10}{2} \][/tex]
[tex]\[ x = \frac{12}{2} \quad \text{or} \quad x = \frac{-8}{2} \][/tex]
[tex]\[ x = 6 \quad \text{or} \quad x = -4 \][/tex]
2. Determine the intervals for the inequality [tex]\( x^2 - 2x - 24 < 0 \)[/tex]:
The roots divide the number line into three intervals: [tex]\( (-\infty, -4) \)[/tex], [tex]\( (-4, 6) \)[/tex], and [tex]\( (6, \infty) \)[/tex]. We test a point in each interval to determine where the inequality holds.
- For [tex]\( x < -4 \)[/tex] (e.g., [tex]\( x = -5 \)[/tex]):
[tex]\[ (-5)^2 - 2(-5) - 24 = 25 + 10 - 24 = 11 > 0 \][/tex]
- For [tex]\( -4 < x < 6 \)[/tex] (e.g., [tex]\( x = 0 \)[/tex]):
[tex]\[ 0^2 - 2(0) - 24 = -24 < 0 \][/tex]
- For [tex]\( x > 6 \)[/tex] (e.g., [tex]\( x = 7 \)[/tex]):
[tex]\[ 7^2 - 2(7) - 24 = 49 - 14 - 24 = 11 > 0 \][/tex]
Therefore, the solution for the first inequality is:
[tex]\[ -4 < x < 6 \][/tex]
### Step 2: Solve the Second Inequality
The second inequality is:
[tex]\[ -x^2 + 9x - 14 \geq 0 \][/tex]
1. Find the roots of the equation [tex]\( -x^2 + 9x - 14 = 0 \)[/tex]:
[tex]\[ -x^2 + 9x - 14 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -14 \)[/tex]:
[tex]\[ x = \frac{-9 \pm \sqrt{9^2 - 4(-1)(-14)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{-9 \pm \sqrt{81 - 56}}{-2} \][/tex]
[tex]\[ x = \frac{-9 \pm \sqrt{25}}{-2} \][/tex]
[tex]\[ x = \frac{-9 \pm 5}{-2} \][/tex]
[tex]\[ x = \frac{-9 + 5}{-2} \quad \text{or} \quad x = \frac{-9 - 5}{-2} \][/tex]
[tex]\[ x = \frac{-4}{-2} \quad \text{or} \quad x = \frac{-14}{-2} \][/tex]
[tex]\[ x = 2 \quad \text{or} \quad x = 7 \][/tex]
2. Determine the intervals for the inequality [tex]\( -x^2 + 9x - 14 \geq 0 \)[/tex]:
The roots divide the number line into three intervals: [tex]\( (-\infty, 2) \)[/tex], [tex]\( (2, 7) \)[/tex], and [tex]\( (7, \infty) \)[/tex]. We test a point in each interval to determine where the inequality holds.
- For [tex]\( x < 2 \)[/tex] (e.g., [tex]\( x = 1 \)[/tex]):
[tex]\[ -(1)^2 + 9(1) - 14 = -1 + 9 - 14 = -6 < 0 \][/tex]
- For [tex]\( 2 < x < 7 \)[/tex] (e.g., [tex]\( x = 3 \)[/tex]):
[tex]\[ -(3)^2 + 9(3) - 14 = -9 + 27 - 14 = 4 \geq 0 \][/tex]
- For [tex]\( x > 7 \)[/tex] (e.g., [tex]\( x = 8 \)[/tex]):
[tex]\[ -(8)^2 + 9(8) - 14 = -64 + 72 - 14 = -6 < 0 \][/tex]
Therefore, the solution for the second inequality is:
[tex]\[ 2 \leq x \leq 7 \][/tex]
### Step 3: Find the Intersection of the Solutions
- First inequality: [tex]\( -4 < x < 6 \)[/tex]
- Second inequality: [tex]\( 2 \leq x \leq 7 \)[/tex]
The intersection of these solutions is:
[tex]\[ 2 \leq x < 6 \][/tex]
### Final Answer
The values of [tex]\( x \)[/tex] which satisfy both inequalities are:
[tex]\[ 2 \leq x < 6 \][/tex]
[tex]\[ \begin{array}{l} x^2-2 x-24<0 \\ -x^2+9 x-14 \geq 0 \end{array} \][/tex]
we will solve each inequality separately and then find their intersection.
### Step 1: Solve the First Inequality
The first inequality is:
[tex]\[ x^2 - 2x - 24 < 0 \][/tex]
1. Find the roots of the equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex]:
[tex]\[ x^2 - 2x - 24 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 96}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{100}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm 10}{2} \][/tex]
[tex]\[ x = \frac{12}{2} \quad \text{or} \quad x = \frac{-8}{2} \][/tex]
[tex]\[ x = 6 \quad \text{or} \quad x = -4 \][/tex]
2. Determine the intervals for the inequality [tex]\( x^2 - 2x - 24 < 0 \)[/tex]:
The roots divide the number line into three intervals: [tex]\( (-\infty, -4) \)[/tex], [tex]\( (-4, 6) \)[/tex], and [tex]\( (6, \infty) \)[/tex]. We test a point in each interval to determine where the inequality holds.
- For [tex]\( x < -4 \)[/tex] (e.g., [tex]\( x = -5 \)[/tex]):
[tex]\[ (-5)^2 - 2(-5) - 24 = 25 + 10 - 24 = 11 > 0 \][/tex]
- For [tex]\( -4 < x < 6 \)[/tex] (e.g., [tex]\( x = 0 \)[/tex]):
[tex]\[ 0^2 - 2(0) - 24 = -24 < 0 \][/tex]
- For [tex]\( x > 6 \)[/tex] (e.g., [tex]\( x = 7 \)[/tex]):
[tex]\[ 7^2 - 2(7) - 24 = 49 - 14 - 24 = 11 > 0 \][/tex]
Therefore, the solution for the first inequality is:
[tex]\[ -4 < x < 6 \][/tex]
### Step 2: Solve the Second Inequality
The second inequality is:
[tex]\[ -x^2 + 9x - 14 \geq 0 \][/tex]
1. Find the roots of the equation [tex]\( -x^2 + 9x - 14 = 0 \)[/tex]:
[tex]\[ -x^2 + 9x - 14 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -14 \)[/tex]:
[tex]\[ x = \frac{-9 \pm \sqrt{9^2 - 4(-1)(-14)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{-9 \pm \sqrt{81 - 56}}{-2} \][/tex]
[tex]\[ x = \frac{-9 \pm \sqrt{25}}{-2} \][/tex]
[tex]\[ x = \frac{-9 \pm 5}{-2} \][/tex]
[tex]\[ x = \frac{-9 + 5}{-2} \quad \text{or} \quad x = \frac{-9 - 5}{-2} \][/tex]
[tex]\[ x = \frac{-4}{-2} \quad \text{or} \quad x = \frac{-14}{-2} \][/tex]
[tex]\[ x = 2 \quad \text{or} \quad x = 7 \][/tex]
2. Determine the intervals for the inequality [tex]\( -x^2 + 9x - 14 \geq 0 \)[/tex]:
The roots divide the number line into three intervals: [tex]\( (-\infty, 2) \)[/tex], [tex]\( (2, 7) \)[/tex], and [tex]\( (7, \infty) \)[/tex]. We test a point in each interval to determine where the inequality holds.
- For [tex]\( x < 2 \)[/tex] (e.g., [tex]\( x = 1 \)[/tex]):
[tex]\[ -(1)^2 + 9(1) - 14 = -1 + 9 - 14 = -6 < 0 \][/tex]
- For [tex]\( 2 < x < 7 \)[/tex] (e.g., [tex]\( x = 3 \)[/tex]):
[tex]\[ -(3)^2 + 9(3) - 14 = -9 + 27 - 14 = 4 \geq 0 \][/tex]
- For [tex]\( x > 7 \)[/tex] (e.g., [tex]\( x = 8 \)[/tex]):
[tex]\[ -(8)^2 + 9(8) - 14 = -64 + 72 - 14 = -6 < 0 \][/tex]
Therefore, the solution for the second inequality is:
[tex]\[ 2 \leq x \leq 7 \][/tex]
### Step 3: Find the Intersection of the Solutions
- First inequality: [tex]\( -4 < x < 6 \)[/tex]
- Second inequality: [tex]\( 2 \leq x \leq 7 \)[/tex]
The intersection of these solutions is:
[tex]\[ 2 \leq x < 6 \][/tex]
### Final Answer
The values of [tex]\( x \)[/tex] which satisfy both inequalities are:
[tex]\[ 2 \leq x < 6 \][/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
