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If "x" is an integer, which of the following inequalities has (have) a finite range of values of "x" satisfying it (them)?
- a)x2 + 5x + 6 > 0
- b)|x + 2| > 4
- c)9x - 7 < 3x + 14
- d)x2 - 4x + 3 < 0
- e)(B) and (D)
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If "x" is an integer, which of the following inequalities ha...
Step 1: Find the values of "x" that will satisfy the four inequalities
Choice A: x2 + 5x + 6 > 0
Factorize the given expression: x2 + 5x + 6 > 0 = (x + 2)(x + 3) > 0.
This inequality will hold good when both (x + 2) and (x + 3) are simultaneously positive OR are simultaneously negative.
Factorize the given expression: x2 + 5x + 6 > 0 = (x + 2)(x + 3) > 0.
This inequality will hold good when both (x + 2) and (x + 3) are simultaneously positive OR are simultaneously negative.
Possibility 1: Both (x + 2) and (x + 3) are positive.
i.e., x + 2 > 0 AND x + 3 > 0
i.e., x > -2 AND x > -3
Essentially translates to x > -2
i.e., x + 2 > 0 AND x + 3 > 0
i.e., x > -2 AND x > -3
Essentially translates to x > -2
Possibility 2: Both (x + 2) and (x + 3) are negative.
i.e., x + 2 < 0 AND x + 3 < 0
i.e., x < -2 AND x < -3
Essentially translates to x < -3
i.e., x + 2 < 0 AND x + 3 < 0
i.e., x < -2 AND x < -3
Essentially translates to x < -3
Evaluating both the possibilities, we get the range of values of "x" that satisfy this inequality to be x > -2 or x < -3. i.e., "x" does not lie between -3 and -2.
i.e., x takes values lesser than -3 or greater than -2.
The range of values that x takes is infinite.
i.e., x takes values lesser than -3 or greater than -2.
The range of values that x takes is infinite.
Choice B: |x + 2| > 4
|x + 2| > 4 is a modulus function and therefore, has two possibilities
|x + 2| > 4 is a modulus function and therefore, has two possibilities
Possiblity 1: x + 2 > 4
i.e., x > 2
i.e., x > 2
Possiblity 2: (x + 2) < -4.
i.e., x < -6
Evaluating the two options together, we get the values of "x" that satisfy the inequality as x > 2 OR x < -6.
i.e., "x" does not lie between -6 and 2.
An infinite range of values.
i.e., x < -6
Evaluating the two options together, we get the values of "x" that satisfy the inequality as x > 2 OR x < -6.
i.e., "x" does not lie between -6 and 2.
An infinite range of values.
Choice C: 9x - 7 < 3x + 14
Simplifying, we get 6x < 21 or x < 3.5.
An infinite range of values.
Simplifying, we get 6x < 21 or x < 3.5.
An infinite range of values.
Choice D: x2 - 4x + 3 < 0
Factorizing x2 - 4x + 3 < 0 we get, (x - 3)(x - 1) < 0.
This inequality will hold good when one of the terms (x - 3) or (x - 1) is positive and the other is negative.
Factorizing x2 - 4x + 3 < 0 we get, (x - 3)(x - 1) < 0.
This inequality will hold good when one of the terms (x - 3) or (x - 1) is positive and the other is negative.
Possibility 1: (x -3) is positive and (x - 1) is negative.
i.e., x - 3 > 0 AND x -1 < 0
i.e., x > 3 AND x < 1
Such a number DOES NOT exist. It is an infeasible solution.
i.e., x - 3 > 0 AND x -1 < 0
i.e., x > 3 AND x < 1
Such a number DOES NOT exist. It is an infeasible solution.
Possibility 2: (x - 3) is negative and (x - 1) is positive.
i.e., x - 3 < 0 AND x - 1 > 0
i.e., x < 3 AND x > 1
Essentially translates to 1 < x < 3 Finite range of values for "x".
i.e., x - 3 < 0 AND x - 1 > 0
i.e., x < 3 AND x > 1
Essentially translates to 1 < x < 3 Finite range of values for "x".
Choice D is the correct answer.
Most Upvoted Answer
If "x" is an integer, which of the following inequalities ha...
Explanation:
Range of values of x satisfying the given inequalities:
- For inequality (b): |x + 2| > 4
- This inequality can be split into two separate inequalities:
- x + 2 > 4
- x + 2 < />
- Solving these inequalities gives x > 2 and x < -6,="" which="" means="" the="" range="" of="" values="" for="" x="" is="" />
- For inequality (d): x^2 - 4x + 3 < />
- This is a quadratic inequality that can be factored as (x - 1)(x - 3) < />
- The solutions to this inequality are 1 < x="" />< 3,="" which="" represents="" a="" finite="" range="" of="" values="" for="" />
Therefore, the correct answer is option 'D' as both inequalities (b) and (d) have a finite range of values of x satisfying them.
Range of values of x satisfying the given inequalities:
- For inequality (b): |x + 2| > 4
- This inequality can be split into two separate inequalities:
- x + 2 > 4
- x + 2 < />
- Solving these inequalities gives x > 2 and x < -6,="" which="" means="" the="" range="" of="" values="" for="" x="" is="" />
- For inequality (d): x^2 - 4x + 3 < />
- This is a quadratic inequality that can be factored as (x - 1)(x - 3) < />
- The solutions to this inequality are 1 < x="" />< 3,="" which="" represents="" a="" finite="" range="" of="" values="" for="" />
Therefore, the correct answer is option 'D' as both inequalities (b) and (d) have a finite range of values of x satisfying them.
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If "x" is an integer, which of the following inequalities has (have) a finite range of values of "x" satisfying it (them)?a)x2+ 5x + 6 > 0b)|x + 2| > 4c)9x - 7 < 3x + 14d)x2- 4x + 3 < 0e)(B) and (D)Correct answer is option 'D'. Can you explain this answer?
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