If |y-1/2| < 11/2, which of the following could be a value

Why is

|y-1/2| < 11/2 equivalent to -11/2 < |y-1/2| < 11/2

Why am I not seeing this and why didn´t I come up with it?

It seems that you need to brush up fundamentals on absolute value.

Theory: math-absolute-value-modulus-86462.html

PS questions on Absolute Values: search.php?search_id=tag&tag_id=58
DS questions on Absolute Values: search.php?search_id=tag&tag_id=37

Tough inequality and absolute value questions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.

Thanks Bunnel for such a valuable reply..and +1 for you as well

I was wondering, if the equation stands at -5<y<6, should not it be that both B & C are right..
Thanks

-11/2=-5.5<-5 and we know that -5<y, thus y cannot be -5.5.

Hope it's clear.

The inequality becomes:
-11/2 < y-1/2 < +11/2

Hence, 1/2 has to be added to each inequality:
-11/2 + 1/2 = -5
11/2 + 1/2 = 6

After simplifying, we get:
-5< y < 6
The only option that lies between -5 and 6 is 11/2, which when simplified is 5.5

Thus, the answer has to be C.

If |y-1/2| < 11/2

First calculate origin (y-1/2)=0. Origin is +1/2.

This should be read as distance from origin 1/2 is less than 11/2 units. So there is range here.

So, on the right side of 1/2 we add 11/2 units and on the left side we minus 11/2 units

(1/2-11/2= -5)---------------------(1/2)---------------------(1/2+11/2 = 6)

So our answer must be in range of -5 to 6. -5<y<6

only 11/2 lies in this range. Hence answer.

Here's a visual solution for this question:

\(|y-\frac{1}{2}|\) represents the distance of y from the point \(\frac{1}{2}\) = 0.5 on the number line.

The given inequality tells us that the distance of y from the point 0.5 is less than \(\frac{11}{2}\) (that is, 5.5)

The points that are at a distance of 5.5 from 0.5 on the number line are: -5 and 6



This means, -5 < y < 6

Therefore, Option C is the correct answer.

If you are confused by the algebraic solution, then it might be easier to plug in when absolute values are involved.

Solution:

We must solve the inequality for two cases:

Case 1: (y-1/2) is Positive:

|y – ½| < 11/2

y – ½ < 11/2

Add ½ to both sides of the equation:

y < 6

Case 2: (y-1/2) is Negative:

|y – ½| < 11/2

-(y – ½) < 11/2

-y + ½ < 11/2

Subtract ½ from both sides of the equation:

-y < 5

y > -5

So we know that y is greater than -5 and less than 6. The only answer choice that fits this criterion is answer choice C, 11/2.

Answer: C

Given: \(|y-\frac{1}{2}| < \frac{11}{2}\)

Asked: Which of the following could be a value of y?

\(|y-\frac{1}{2}| < \frac{11}{2}\)
\(-\frac{11}{2}<y-\frac{1}{2}<\frac{11}{2}\)
\(-\frac{11}{2}+\frac{1}{2}<y<\frac{11}{2}+\frac{1}{2}\)
\(-5<y<6\)

Since 11/2 = 5.5 lies within the range \(-5<y<6\)

IMO C

This is a type 1 variable single expression with the mod on one side and a numerical value on the other side of the inequality.
Y -1/2 <11/2
Y <6

Y - 1/2 > -11/2 ( reverse the inequality rêverse the sign)
Y>-5
So combining the inequalities we get -5<y<6
Only c fits
C

It's definitely not necessary to work out the algebra or to test all the values.

First, notice that the expression inside the absolute value is only 1/2 away from y. In other words, if we plug in a number that is much bigger than 11/2, either + or -, there is no way the absolute value can end up less than 11/2. If you happen to test one of A, D, or E, you should recognize that your result is far too big. Definitely don't test the rest!

That leaves B and C. These might seem to be the same, since we're dealing with absolute value. However, a good principle to bear in mind is that when the signs of terms inside an absolute value are opposite, they work against each other to make the absolute value smaller, but when the signs are the same, they work together to make the absolute value larger. In this case, |11/2 − 1/2| = 10/2 = 5, while |-11/2 − 1/2| = |-12/2| = |-6| = 6. Too big!

If the principle above didn't jump out to you, you could of course just test B and C. I'd start with the positive version, since it's simpler. Since our y value is much larger than 1/2, we can ignore the absolute value entirely. We just have to agree that 11/2 - 1/2 < 11/2, since anything gets smaller when you take away from it!

, which of the following could be a value of y?

(A) -11
(B) -11/2
(C) 11/2
(D) 11
(E) 22