IDNLearn.com is your go-to platform for finding accurate and reliable answers. Ask any question and get a detailed, reliable answer from our 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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.
