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Difficulty: Sub 505 Level,   Absolute Values,   Inequalities,                           
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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Flexxice wrote:
Attachment:
Math question.jpg

On the number line, the shaded interval is the graph of which of the following inequalities?

A) |x| <= 4

B) |x| <= 8

C) |x - 2| <= 4

D) |x - 2| <= 6

E) |x + 2| <= 6

Since the question appears in the Official Guide, I can see the GMAC's solution.

However, I barely have any knowledge about graphing inequalities. I know when to fill in and when not to fill in the dot and can graph a very easy inequality like x < 4, but this question is by far too much for me. For example, the solution says something about the midpoint of the interval from -8 to 4. Of course I can calculate the midpoint of that, but why is this the interval? And how do you go on with the calculation? I have never covered this at school.

Could someone please write a short guide about how to solve graphing inequalities problems? What one has to do and how you do it. I cannot find anything like that on the internet :(

Thanks a lot for any answer!


Hi,
you can easily discard A, B and C as the total in each choice can never be 8 or 4, which are the edge points of the LINE...
For choosing between D and E, let me tell you what does a linear equality actually mean..
1) |x|<=4
this is SAME as |x-0|<=4..
this means x is less than or equal to 4 units away from 0 on either side.. so edge points are 0+4 on ONE end and 0-4 on the OTHER end...

2) |x - 1|<=4
this means x is less than or equal to 4 units away from 1 on either side.. so edge points are 1+4=5 on ONE end and 1-4=-3 on the OTHER end

similarly try for D and E, and you will realize E is the answer
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
Thanks a lot to all of you! You guys really helped me a lot!

One last question:

If it asked about graphing /x + 2/ => 6, what would the illustration look like?
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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Flexxice wrote:
Thanks a lot to all of you! You guys really helped me a lot!

One last question:

If it asked about graphing /x + 2/ => 6, what would the illustration look like?


The shading would be the reverse of current shading in that case. Whatever is shaded now would be unshaded and whatever is not shaded right now, would be shaded.
In that case, the shaded region would be 4 and to the right of 4 and -8 and to the left of -8. All these values will satisfy |x+2| >= 6
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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Hi All,

You can use the answer choices 'against' the prompt to quickly eliminate options and zero-in on the correct answer.

From the given Number Line, we know that every number form -8 to +4 (inclusive) must 'fit' the correct answer, so we can start by plugging -8 and +4 into each of the choices. If you find an answer is not mathematically correct, then you can eliminate it. By plugging in these two values, we can quickly eliminate Answers A, C and D.

From here, we have to 'nitpick' what each inequality actually refers to in a bit more detail. Answer B tells us that |X| <= 8, but that Number Line would include X=5, X=6, X=7 and X=8 among it's solutions.... and the Number Line we were given does NOT. Thus, we can eliminate Answer B.

Final Answer:

GMAT assassins aren't born, they're made,
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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On the number line, the shaded interval is the graph of which of the following inequalities?

A) |x| <= 4

B) |x| <= 8

C) |x - 2| <= 4

D) |x - 2| <= 6

E) |x + 2| <= 6

Explanation:

Case1: for x+2>=0 i.e for x>=(-2)
We can expand modulus as,
x+2<=6
x<=4

Case2: for x+2<0 i.e for x<(-2)
We can expand modulus as,
-(x+2)<=6
x>=(-8)

Combine case 1 & case 2 to get the overall range
-8<=x<=4

Hence E is the correct answer.

Award kudos if you like the explanation.?

Regards,
Atul Pandey

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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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Flexxice wrote:
Attachment:
line.jpg


On the number line, the shaded interval is the graph of which of the following inequalities?

A) \(|x| \leq 4\)

B) \(|x| \leq 8\)

C) \(|x - 2| \leq 4\)

D) \(|x - 2| \leq 6\)

E) \(|x + 2| \leq 6\)


Solution:

Recall that |x - c| = d (where d is positive) means x is exactly d units from c. Therefore, |x - c| ≤ d (where d is positive) means all the values of x that are d units or less from c. For example, let’s analyze choice A, which can be expressed as |x - 0| ≤ 4. If this is the correct answer, then the solution set consists of all the numbers that are 4 units or less from 0, which means the darkened interval on the number line should be from -4 to 4, inclusive. However, that is not what the given graph shows. So A is not the correct answer.

Let’s analyze choice E, which can be expressed as |x - (-2)| ≤ 6. If this is the correct answer, then the solution set consists of all the numbers that are 6 units or less from -2, which means the darkened interval on the number line should be from -8 to 4, inclusive. Since this is exactly what the given graph shows, then E is the correct answer.

(Note: A shortcut to solve this problem using the given graph (and provided that we know the meaning of |x - c| ≤ d) is: The value of c is the midpoint of the two endpoints of the interval and d is the distance between one of the endpoints and c. For example, here the two endpoints of the interval are -8 and 4, so c = (-8 + 4)/2 = -2 and d = |4 - (-2)| = |-8 - (-2)| = 6. Therefore, the correct inequality is: |x - (-2)| ≤ 6, i.e., |x + 2| ≤ 6.)

Answer: E
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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The midpoint of the number line is -2. Lets take a look at the choices:

A) \(|x| \leq 4\) -- quickly eliminate

B) \(|x| \leq 8\) -- quickly eliminate

C) \(|x - 2| \leq 4\)

\( -4 \leq x - 2 \leq 4\)
\( -2 \leq x \leq 6\) -- Incorrect

D) \(|x - 2| \leq 6\)
\( -6 \leq x - 2 \leq 6\)
\( -4 \leq x \leq 8\) -- Incorrect

E) \(|x + 2| \leq 6\)
\( -6 \leq x + 2 \leq 6\)
\( -8 \leq x \leq 4\) -- CORRECT

Answer is E.
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On the number line, the shaded interval is the graph of which of the [#permalink]
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Flexxice wrote:
Attachment:
line.jpg


On the number line, the shaded interval is the graph of which of the following inequalities?

A) \(|x| \leq 4\)

B) \(|x| \leq 8\)

C) \(|x - 2| \leq 4\)

D) \(|x - 2| \leq 6\)

E) \(|x + 2| \leq 6\)

Since the question appears in the Official Guide, I can see the GMAC's solution.

However, I barely have any knowledge about graphing inequalities. I know when to fill in and when not to fill in the dot and can graph a very easy inequality like x < 4, but this question is by far too much for me. For example, the solution says something about the midpoint of the interval from -8 to 4. Of course I can calculate the midpoint of that, but why is this the interval? And how do you go on with the calculation? I have never covered this at school.

Could someone please write a short guide about how to solve graphing inequalities problems? What one has to do and how you do it. I cannot find anything like that on the internet :(

Thanks a lot for any answer!


The in value is on the number line is expressed as: \( -8 \leq x \leq 4\)

Now, we need to add two sides and divide the result by 2: \(-8+4=\frac{-4}{2}=-2\)

Now, subtracting the \(-2\) from every element of equality : \( -8-(-2) \leq x-(-2) \leq 4-(-2)\)

= \( -6 \leq x + 2\leq 6\), which is \(|x + 2| \leq 6\)

The answer is E
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Re: On the number line, the shaded interval is the graph of which of the [#permalink]
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Flexxice wrote:
Attachment:
line.jpg


On the number line, the shaded interval is the graph of which of the following inequalities?

A) |x| ≤ 4
B) |x| ≤ 8
C) |x - 2| ≤ 4
D) |x - 2| ≤ 6
E) |x + 2| ≤ 6



STRATEGY: Upon reading any GMAT Problem Solving question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test x-values that are included in the shaded interval.
Now let's give ourselves up to 20 seconds to identify a faster approach.
Another approach is to solve each answer choice for x until we find one whose solution matches the given shaded interval.
I'm pretty sure testing values is going to be a lot faster and much easier


When we check the number line, we can see that x = -8 is included in the shaded interval, which means x = -8 must also be a solution to the correct answer choice.

Now plug x = -8 into each answer choice to get:
A) |-8| ≤ 4. This simplifies to 8 ≤ 4, which is not true. ELIMINATE.
B) |-8| ≤ 8. This simplifies to 8 ≤ 8, which is true. KEEP.
C) |(-8) - 2| ≤ 4. This simplifies to 10 ≤ 4, which is not true. ELIMINATE.
D) |(-8) - 2| ≤ 6. This simplifies to 10 ≤ 6, which is not true. ELIMINATE.
E) |(-8) + 2| ≤ 6. This simplifies to 6 ≤ 6, which is true. KEEP.
We're already down to just two answer choices!

Let's test another extreme x-value...
We can see that x = 4 is also included in the shaded interval, which means x = 4 must also be a solution to the correct answer choice.
Plug x = 4 into the two remaining choices to get:
B) |4| ≤ 8. This simplifies to 4 ≤ 8, which is true. KEEP.
E) |4 + 2| ≤ 6. This simplifies to 6 ≤ 6, which is true. KEEP.
That was no help!

When we examine the two remaining answer choices (B and E), we can see that choice B tells us x CAN equal 8 (since x = 8 satisfies the inequality |x| ≤ 8) , whereas choice E tells us x CAN'T equal 8 (since x = 8 does not satisfy the inequality |x + 2| ≤ 6).

When we check the number line, we can see that x = 8 is NOT included in the shaded region of the number line.
In other words x = 8 cannot be a solution, which means we can eliminate choice B, because it tells us that x CAN equal 8.

By the process of elimination, the correct answer is E.
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