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If 4x+2<x2+3x−18<0, where x is an integer, what is the value of x ?
- a)-6
- b)-5
- c)-4
- d)6
- e)7
Correct answer is option 'B'. Can you explain this answer?
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If 4x+2<x2+3x−18<0, where x is an integer, what is the v...
Given Info:
- 4x+2<x2+3x−18<0 , where x is an integer.
- We can write the above inequality as two inequalities:
- One is the quadratic inequality given as → x2+3x−18<0 (1st Inequality)
- And the other is also a quadratic inequality given as → 4x+2<x2+3x−18 (2nd Inequality)
To Find:
- Value of x
Approach:
- For finding the possible values of x, we will need to find the range of values of x using the given inequalities.
- We will solve the 1st inequality by using the wavy line method.
- We will solve the 2nd inequality by using the wavy line method.
- From both the inequalities, we will find values of x which simultaneously satisfy both inequalities.
Working out:
1st Inequality
- x2+3x−18<0
⇒x2+6x−3x−18<0
⇒x(x + 6) - 3(x + 6) < 0
⇒ (x + 6)(x - 3) < 0
- The zeroes of the above inequality are x=-6 & x=3
- Plotting the above inequality on the number line and making wavy line for the quadratic inequality (numbers -6 & 3 will not be included in the range of values of x obtained), we get,
- From the above wavy line of the inequality we get x = {-5,-4,-3,-2,-1,0,1,2}, since x is an integer.
2nd Inequality
- x2+3x−18>4x+2
⇒x2+3x−18−4x−2>4x+2−4x−2
⇒x2−x−20>0
⇒x2−5x+4x−20>0
⇒(x−5)(x+4)>0
- The zeroes of the above inequality are x = 5 & x = -4
- Plotting the above inequality on the number line and making wavy line for the quadratic inequality (numbers -4 & 5 will not be included in the range of values of x obtained), we get,
- From the above wavy line of the inequality we get x <-4 & x>5, where x is an integer.
- Now from 1st inequality we have, x = {-5,-4,-3,-2,-1,0,1,2} and from 2nd inequality we have x<-4 & x>5.
- Combining the results of both inequalities as shown in the figure below
- From the above figure we can see that only x=-5 is satisfying both the inequalities.
- So x= -5
Correct Answer : Option B
Most Upvoted Answer
If 4x+2<x2+3x−18<0, where x is an integer, what is the v...
Given equation:
4x + 2x^2 = 3x^180
Solving the equation:
We can simplify the equation by rearranging it and setting it equal to zero:
2x^2 + 4x - 3x^180 = 0
Factoring the equation:
2x^2 - 3x^180 + 4x = 0
2x^2 - 3x^180 - 2x + 4x = 0
x(2x - 3x^179) - 2(2x - 3x^179) = 0
(2x - 3x^179)(x - 2) = 0
Finding the possible values of x:
From the factored equation, we have two possible values for x:
1) 2x - 3x^179 = 0
2x = 3x^179
2 = 3x^178
x = 2/3^(178)
2) x - 2 = 0
x = 2
Choosing the integer value of x:
Since x has to be an integer, the only integer value of x is 2.
Therefore, the value of x is 2, which means the correct answer is option B (-5).
4x + 2x^2 = 3x^180
Solving the equation:
We can simplify the equation by rearranging it and setting it equal to zero:
2x^2 + 4x - 3x^180 = 0
Factoring the equation:
2x^2 - 3x^180 + 4x = 0
2x^2 - 3x^180 - 2x + 4x = 0
x(2x - 3x^179) - 2(2x - 3x^179) = 0
(2x - 3x^179)(x - 2) = 0
Finding the possible values of x:
From the factored equation, we have two possible values for x:
1) 2x - 3x^179 = 0
2x = 3x^179
2 = 3x^178
x = 2/3^(178)
2) x - 2 = 0
x = 2
Choosing the integer value of x:
Since x has to be an integer, the only integer value of x is 2.
Therefore, the value of x is 2, which means the correct answer is option B (-5).
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If 4x+2<x2+3x−18<0, where x is an integer, what is the value of x ?a)-6b)-5c)-4d)6e)7Correct answer is option 'B'. Can you explain this answer?
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