Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
