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e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3134
How many non-negative integers satisfy the inequality x^2 – 7x – 18 .  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 48% (02:03) correct 52% (02:11) wrong based on 86 sessions

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How many non-negative integers satisfy the inequality $$\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0$$?

A. 6
B. 7
C. 8
D. 9
E. Infinite

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How many non-negative integers satisfy the inequality x^2 – 7x – 18 .  [#permalink]

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$$(x2–7x–18)(x+5)/(x2–4)≤0$$

= $$((x+2)(x-9)(x+5))/((x-2)(x+2))$$
= ((x-9)(x+5))/(x-2))
non-negative implies 0,1,2,3,4.....

when x = 0,1 -ve sign cancels
at x= 2 denominator is zero implies the value is infinty. So, x!= 2
x = 3,4,5,6,7,8,9
numerator is negative OR Zero and denominator is positive
Rest of the values x >9 given equation is positive
So, Given equation satisfies, implies for 7 integers it satisfies the given condition

Option B is correct

Originally posted by Dare Devil on 16 Jan 2019, 02:25.
Last edited by Dare Devil on 18 Jan 2019, 08:31, edited 1 time in total.
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How many non-negative integers satisfy the inequality x^2 – 7x – 18 .  [#permalink]

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simplify relation we get
(x-9) * ( x+5)/ ( x-2) <=0

values of x would satisfy relation <=0
x at 2,3,4,5,6,7,8,9 satisfies the relation
as for 0,1, we get >0 values
and for integers beyond 9 as well we get >0 values

IMO C ; 8

EgmatQuantExpert wrote:
How many non-negative integers satisfy the inequality $$\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0$$?

A. 6
B. 7
C. 8
D. 9
E. Infinite

e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3134
Re: How many non-negative integers satisfy the inequality x^2 – 7x – 18 .  [#permalink]

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1

Solution

Given:
• An inequality, $$\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0$$

To find:
• The number of non-negative integer values of x that satisfy the given inequality

Approach and Working:
• $$\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0$$

Let’s factorise the quadratic expressions in the numerator and denominator
• $$\frac{(x – 9)(x + 2)(x + 5)}{(x + 2)(x - 2)} ≤ 0$$

We can cancel out the common term, x + 2, in both numerator and denominator
• $$\frac{(x – 9)(x + 5)}{(x - 2)} ≤ 0$$

Now, multiplying and dividing by (x – 2), we get,
• $$\frac{(x – 2)(x - 9)(x + 5)}{(x - 2)^2} ≤ 0$$

In the above expression, the denominator is always greater than 0. Thus, the numerator must be ≤ 0.
• (x – 2)(x - 9)(x + 5) ≤ 0

Representing the zero points {-5, 2, 9} on a number line, we can find the regions in which the expression is ≤ 0 • From the above figure, we can see that (x – 2)(x - 9)(x + 5) ≤ 0 in the regions, 2 ≤ x ≤ 9 and x ≤ -5
• And, we know that x ≠ 2, since, the denominator of the given expression cannot be = 0

We are asked to find the number of non-negative integer values of x.
• Therefore, the values of x can be {3, 4, 5 ,6 ,7 ,8, 9}

Hence the correct answer is Option B.

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Re: How many non-negative integers satisfy the inequality x^2 – 7x – 18 .  [#permalink]

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EgmatQuantExpert wrote:

Solution

Given:
• An inequality, $$\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0$$

To find:
• The number of non-negative integer values of x that satisfy the given inequality

Approach and Working:
• $$\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0$$

Let’s factorise the quadratic expressions in the numerator and denominator
• $$\frac{(x – 9)(x + 2)(x + 5)}{(x + 2)(x - 2)} ≤ 0$$

We can cancel out the common term, x + 2, in both numerator and denominator
• $$\frac{(x – 9)(x + 5)}{(x - 2)} ≤ 0$$

Now, multiplying and dividing by (x – 2), we get,
• $$\frac{(x – 2)(x - 9)(x + 5)}{(x - 2)^2} ≤ 0$$

In the above expression, the denominator is always greater than 0. Thus, the numerator must be ≤ 0.
• (x – 2)(x - 9)(x + 5) ≤ 0

Representing the zero points {-5, 2, 9} on a number line, we can find the regions in which the expression is ≤ 0 • From the above figure, we can see that (x – 2)(x - 9)(x + 5) ≤ 0 in the regions, 2 ≤ x ≤ 9 and x ≤ -5
• And, we know that x ≠ 2, since, the denominator of the given expression cannot be = 0

We are asked to find the number of non-negative integer values of x.
• Therefore, the values of x can be {3, 4, 5 ,6 ,7 ,8, 9}

Hence the correct answer is Option B.

@EgmatQuantExpert"

the expression
[m]\frac{(x^2 – 7x – 18)(x + 5)}{(x^2 – 4)} ≤ 0

so wont it be =0 and should be valid at x=2 would be a valid to count as an integer ...
considering this relation only I opted for option C '8' over 7 option b .. Re: How many non-negative integers satisfy the inequality x^2 – 7x – 18 .   [#permalink] 18 Jan 2019, 08:13
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# How many non-negative integers satisfy the inequality x^2 – 7x – 18 .  