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# How many integers x satisfy |2x + 3 | < 6 ?

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Math Expert
Joined: 02 Sep 2009
Posts: 49496
How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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16 Jul 2018, 20:54
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15% (low)

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74% (01:15) correct 26% (01:02) wrong based on 141 sessions

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How many integers x satisfy |2x + 3 | < 6 ?

A. 1

B. 2

C. 6

D. 7

E. 8

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Joined: 02 Aug 2009
Posts: 6810
Re: How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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16 Jul 2018, 22:57
1
1
Bunuel wrote:
How many integers x satisfy |2x + 3 | < 6 ?

A. 1

B. 2

C. 6

D. 7

E. 8

two cases

1) 2x+3 >0
so $$2x+3<6..............2x<3.............x<\frac{3}{2}$$
2) 2x+3<0
so $$2x+3>-6..............2x>-9.............x>\frac{-9}{2}$$

integers value-
-4, -3, -2, -1, 0, 1 ------ 6 values

C
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Re: How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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16 Jul 2018, 23:16
1
1
Bunuel wrote:
How many integers x satisfy |2x + 3 | < 6 ?

A. 1

B. 2

C. 6

D. 7

E. 8

Given, $$|2x + 3 | < 6$$
Or, $$|x-(\frac{-3}{2})|< 3$$ (Dividing both sides of the inequality by 2)
Or, $$(\frac{-3}{2}-3) < x < (\frac{-3}{2}+3)$$
Or, -$$4.5<x<1.5$$
So, Possible integer values of x: -4, -3, -2, -1, 0, 1

So, no of integers are 6.

Ans. (C)
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Re: How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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16 Jul 2018, 23:32
1
1
+1 for C.

|2x+3|<6
Possible cases :
Case 1 = -(2x + 3) <6 = -2x - 3 <6 = -2x < 9 = x > -9/2 = x > -4.5
Case 2 -> 2x + 3 <6 = 2x < 3 = x < 3 / 2 = x < 1.5

Combining :
4.5 < x < 1.5

Therefore,
x = -4, -3, -2, -1, 0, 1

Number of integers = 6.

Hence, C.
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How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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17 Jul 2018, 04:55
Bunuel wrote:
How many integers x satisfy |2x + 3 | < 6 ?

A. 1

B. 2

C. 6

D. 7

E. 8

As this problem is on absolute value , we must have a certain range of values.

Case 1 : 2x + 3 = positive.

2x + 3 < 6

x < 1.5

case 2: 2x +3 = negative

-2x - 3 <6

-2x < 9

x < -4.5.

So, -4.5 < x < 1.5

Integers fall into the range : 1, 0 , -1, -2 , -3 , -4

There are 6 values in total.

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Re: How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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19 Jul 2018, 06:49
Bunuel wrote:
How many integers x satisfy |2x + 3 | < 6 ?

A. 1

B. 2

C. 6

D. 7

E. 8

$$|2x + 3 | < 6$$

$$-6 < 2x + 3 < 6$$

-$$3 < 2x < 9$$

$$-1.5 < x < 4.5$$

Therefore, -1, 0, 1, 2, 3, 4 are the required values.

So C
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Re: How many integers x satisfy |2x + 3 | < 6 ?  [#permalink]

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19 Jul 2018, 17:19
Bunuel wrote:
How many integers x satisfy |2x + 3 | < 6 ?

A. 1

B. 2

C. 6

D. 7

E. 8

Recall that if k is a positive constant and |ax + b| < k, then -k < ax + b < k. So we can solve the given inequality by removing the absolute value sign and change it to the following double inequality:

-6 < 2x + 3 < 6

-9 < 2x < 3

-9/2 < x < 3/2

-4.5 < x < 1.5

The integers x can be are: -4, -3, -2, -1, 0, and 1. So there are 6 integers that satisfy the inequality.

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Re: How many integers x satisfy |2x + 3 | < 6 ? &nbs [#permalink] 19 Jul 2018, 17:19
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