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For how many positive integer values of x, the inequality holds true?
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20 Mar 2023, 11:17
20 Mar 2023, 11:17
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For how many positive integer values of x, the inequality \(\dfrac{(x-1)^2(x-3)}{(x-6)}<0\) holds true?
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Re: For how many positive integer values of x, the inequality holds true?
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08 Apr 2023, 04:28
08 Apr 2023, 04:28
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Expert Reply
To find the number of positive integer values of x for which the inequality [(x−1)^2(x−3)]/(x−6)<0 holds true, we need to analyze the sign of the expression [(x−1)^2(x−3)]/(x−6) for different intervals of x.
First, let's find the values of x for which the expression [(x−1)^2(x−3)]/(x−6) is equal to zero or undefined, which are the points of interest:
(x−1)^2 = 0 when x = 1
(x−3) = 0 when x = 3
(x−6) = 0 when x = 6 (This makes the expression undefined as it results in division by zero.)
So, the points of interest are x = 1, 3, and 6.
Now, let's analyze the sign of the expression [(x−1)^2(x−3)]/(x−6) for different intervals of x, using test points within each interval:
Interval 1: x < 1
Let's choose x = 0 as a test point. Plugging in x = 0 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(0−1)^2(0−3)]/(0−6) = (1)(-3)/(-6) = 1/2 > 0
Interval 2: 1 < x < 3
Let's choose x = 2 as a test point. Plugging in x = 2 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(2−1)^2(2−3)]/(2−6) = (1)(-1)/(-4) = 1/4 > 0
Interval 3: 3 < x < 6
Let's choose x = 4 as a test point. Plugging in x = 4 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(4−1)^2(4−3)]/(4−6) = (9)(1)/(-2) = -9/2 < 0
Interval 4: x > 6
Let's choose x = 7 as a test point. Plugging in x = 7 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(7−1)^2(7−3)]/(7−6) = (36)(4)/(1) = 144 > 0
Based on the analysis above, the expression [(x−1)^2(x−3)]/(x−6) is negative (less than 0) in the interval 3 < x < 6.
So, the number of positive integer values of x for which the inequality [(x−1)^2(x−3)]/(x−6)<0 holds true is 2 (since x can only take integer values within the interval 3 < x < 6).
Therefore, the correct answer is: 2.
First, let's find the values of x for which the expression [(x−1)^2(x−3)]/(x−6) is equal to zero or undefined, which are the points of interest:
(x−1)^2 = 0 when x = 1
(x−3) = 0 when x = 3
(x−6) = 0 when x = 6 (This makes the expression undefined as it results in division by zero.)
So, the points of interest are x = 1, 3, and 6.
Now, let's analyze the sign of the expression [(x−1)^2(x−3)]/(x−6) for different intervals of x, using test points within each interval:
Interval 1: x < 1
Let's choose x = 0 as a test point. Plugging in x = 0 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(0−1)^2(0−3)]/(0−6) = (1)(-3)/(-6) = 1/2 > 0
Interval 2: 1 < x < 3
Let's choose x = 2 as a test point. Plugging in x = 2 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(2−1)^2(2−3)]/(2−6) = (1)(-1)/(-4) = 1/4 > 0
Interval 3: 3 < x < 6
Let's choose x = 4 as a test point. Plugging in x = 4 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(4−1)^2(4−3)]/(4−6) = (9)(1)/(-2) = -9/2 < 0
Interval 4: x > 6
Let's choose x = 7 as a test point. Plugging in x = 7 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(7−1)^2(7−3)]/(7−6) = (36)(4)/(1) = 144 > 0
Based on the analysis above, the expression [(x−1)^2(x−3)]/(x−6) is negative (less than 0) in the interval 3 < x < 6.
So, the number of positive integer values of x for which the inequality [(x−1)^2(x−3)]/(x−6)<0 holds true is 2 (since x can only take integer values within the interval 3 < x < 6).
Therefore, the correct answer is: 2.
Re: For how many positive integer values of x, the inequality holds true?
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07 Oct 2023, 08:31
07 Oct 2023, 08:31
GeminiHeat wrote:
To find the number of positive integer values of x for which the inequality [(x−1)^2(x−3)]/(x−6)<0 holds true, we need to analyze the sign of the expression [(x−1)^2(x−3)]/(x−6) for different intervals of x.
First, let's find the values of x for which the expression [(x−1)^2(x−3)]/(x−6) is equal to zero or undefined, which are the points of interest:
(x−1)^2 = 0 when x = 1
(x−3) = 0 when x = 3
(x−6) = 0 when x = 6 (This makes the expression undefined as it results in division by zero.)
So, the points of interest are x = 1, 3, and 6.
Now, let's analyze the sign of the expression [(x−1)^2(x−3)]/(x−6) for different intervals of x, using test points within each interval:
Interval 1: x < 1
Let's choose x = 0 as a test point. Plugging in x = 0 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(0−1)^2(0−3)]/(0−6) = (1)(-3)/(-6) = 1/2 > 0
Interval 2: 1 < x < 3
Let's choose x = 2 as a test point. Plugging in x = 2 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(2−1)^2(2−3)]/(2−6) = (1)(-1)/(-4) = 1/4 > 0
Interval 3: 3 < x < 6
Let's choose x = 4 as a test point. Plugging in x = 4 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(4−1)^2(4−3)]/(4−6) = (9)(1)/(-2) = -9/2 < 0
Interval 4: x > 6
Let's choose x = 7 as a test point. Plugging in x = 7 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(7−1)^2(7−3)]/(7−6) = (36)(4)/(1) = 144 > 0
Based on the analysis above, the expression [(x−1)^2(x−3)]/(x−6) is negative (less than 0) in the interval 3 < x < 6.
So, the number of positive integer values of x for which the inequality [(x−1)^2(x−3)]/(x−6)<0 holds true is 2 (since x can only take integer values within the interval 3 < x < 6).
Therefore, the correct answer is: 2.
First, let's find the values of x for which the expression [(x−1)^2(x−3)]/(x−6) is equal to zero or undefined, which are the points of interest:
(x−1)^2 = 0 when x = 1
(x−3) = 0 when x = 3
(x−6) = 0 when x = 6 (This makes the expression undefined as it results in division by zero.)
So, the points of interest are x = 1, 3, and 6.
Now, let's analyze the sign of the expression [(x−1)^2(x−3)]/(x−6) for different intervals of x, using test points within each interval:
Interval 1: x < 1
Let's choose x = 0 as a test point. Plugging in x = 0 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(0−1)^2(0−3)]/(0−6) = (1)(-3)/(-6) = 1/2 > 0
Interval 2: 1 < x < 3
Let's choose x = 2 as a test point. Plugging in x = 2 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(2−1)^2(2−3)]/(2−6) = (1)(-1)/(-4) = 1/4 > 0
Interval 3: 3 < x < 6
Let's choose x = 4 as a test point. Plugging in x = 4 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(4−1)^2(4−3)]/(4−6) = (9)(1)/(-2) = -9/2 < 0
Interval 4: x > 6
Let's choose x = 7 as a test point. Plugging in x = 7 into the expression [(x−1)^2(x−3)]/(x−6), we get:
[(7−1)^2(7−3)]/(7−6) = (36)(4)/(1) = 144 > 0
Based on the analysis above, the expression [(x−1)^2(x−3)]/(x−6) is negative (less than 0) in the interval 3 < x < 6.
So, the number of positive integer values of x for which the inequality [(x−1)^2(x−3)]/(x−6)<0 holds true is 2 (since x can only take integer values within the interval 3 < x < 6).
Therefore, the correct answer is: 2.
This is the correct mathematical approach. And could one make an educated guess that the answer will be 2 since the highest power of x in the given inequality = 2, so there may be 2 roots for x?
