What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9

Complete step-by-step solution

What is the value of \(|x + 5| + |x - 3|\) ?

The critical points (aka key points or transition points) are -5 and 3 (the values of x for which the expressions in the absolute values become 0).

Consider three ranges:

• If \(x < - 5\), then \(x + 5 < 0\) and \(x - 3 < 0\), so \(|x + 5| = -(x + 5)\) and \(|x - 3| = -(x - 3)\). Thus in this range \(|x + 5| + |x - 3|\) becomes \(-(x + 5) - (x - 3) = -2 - 2x\).
  
• If \(- 5 \leq x \leq 3\), then \(x + 5 \geq 0\) and \(x - 3 \leq 0\), so \(|x + 5| = x + 5\) and \(|x - 3| = -(x - 3)\). Thus in this range \(|x + 5| + |x - 3|\) becomes \(x + 5 - (x - 3) = 8\).
  
• If \(x > 3\), then \(x + 5 > 0\) and \(x - 3 > 0\), so \(|x + 5| = x + 5\) and \(|x - 3| = x - 3\). Thus in this range \(|x + 5| + |x - 3|\) becomes \(x + 5 + (x - 3) = 2x + 2\).

The above mean that if x is in the first range (\(x < - 5\)) or in the third range (\(x > 3\)), then the value of \(|x + 5| + |x - 3|\) depends on the value of x. For example:

If \(x = -10\), then \(|x + 5| + |x - 3|= -2 - 2x=18\);
If \(x = -7\), then \(|x + 5| + |x - 3|= -2 - 2x=12\);
If \(x = 4\), then \(|x + 5| + |x - 3|= 2x + 2=10\);
If \(x = 6\), then \(|x + 5| + |x - 3|= 2x + 2=14\).

But if x is in the second range (\(- 5 \leq x \leq 3\)), then the value of \(|x + 5| + |x - 3|\) is independent of the value of x, and is ALWAYS equals to 8. For example:

If \(x = -5\), then \(|x + 5| + |x - 3|= 8\);
If \(x = 0\), then \(|x + 5| + |x - 3|= 8\);
If \(x = 3\), then \(|x + 5| + |x - 3|= 8\).

(1) \(x^2< 25\):

Take the square root: \(|x| < 5\);
Get rid of the absolute value sign: \(-5 < 0 < 5\);
x can be in the second or third range. So, \(|x + 5| + |x - 3|\) is either 8 or \(2x + 2\). Not sufficient.

(2) \(x^2 > 9\):

Take the square root: \(|x| > 3\);
Get rid of the absolute value sign: \(x < -3\) or \(x > 3\);
x can be in any of the three ranges from above. So, \(|x + 5| + |x - 3|\) is \(-2 - 2x\), 8 or \(2x + 2\). Not sufficient.


(1)+(2) We get \(-5 < x < -3\) (second range) or \(3 < x < 5\) (third range). If \(-5 < x < -3\) (second range), then \(|x + 5| + |x - 3|=8\) but if \(3 < x < 5\) (third range), then \(|x + 5| + |x - 3|=2x + 2\) (so the value will depend on the exact value of x). Not sufficient.


Answer: E.

So, at first glance I thought it was E, but when combining the inequalities I got -2 < x < 2, which all of the values -1, 0 and 1 cause the equation to equal 8 -- but the OA is E - so what did I do wrong here? Thanks!!

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of |x+5| + |x-3| ?

1) x^2 < 25

2) x^2 > 9


When you modify the original condition and the question, a case where sum of 2 absolute values is derived is that the range of in between gets a consistent answer, which is -5<=x<=3?.
There is 1 variable(x), which should match with the number of equations. so you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
When it comes to inequality questions, if range of que includes range of con, use the fact that that con is sufficient.
For 1), in -5<x<5, the range of que doesn't include the range of con, which is not sufficient.
For 2), in x<-3 or 3<x, the range of que doesn't include the range of con, which is not sufficient.
When 1) & 2), in -5<x<-3 or 3<x<5, the range of que doesn't include the range of con, which is not sufficient.
Thus, the answer is E.


 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

we have three concerned ranges

x < -3 here equation will be -2x-2
-3 =< x < 5 here equation value is 8
x >= 5 Here equation value will be 2x + 2

so only for 2nd range we have a fixed value.

Stmt 1 implies -5 < x < 5 insufficient as it covers more than one of the three above mentioned ranges.
Stmt 2 implies x < -3 and x > 3 insufficient as it covers more than one of the three above mentioned ranges.

Combining statement 1 & 2 -5<x < -3 and 3<x<5 insufficient as it covers more than one of the three above mentioned ranges.

Ans E

Bunuel, when we simplify x^2<25, do we write |x|<5 or x<|5| ?

Thanks

\(x^2 < 25\)

Take the square root from both sides: \(|x| < 5\) (recall that \(\sqrt{x^2}=|x|\)).

Hope it's clear.

An eye opener...
I had missed it.



Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app

Asked: What is the value of |x+5| + |x-3| ?

1) \(x^2\) < 25
|x| < 5
-5<x<5
For the region -5<x<=3
|x+5| + |x-3| = 8
But for the region 3<x<5
|x+5| + |x-3| = 8 + 2(x-3) = 2+2x
NOT SUFFICIENT


2) \(x^2\) > 9
|x| > 3
x<-3 or x>3
|x+5| + |x-3| varies with the value of x
NOT SUFFICIENT

(1) + (2)
1) \(x^2\) < 25
2) \(x^2\) > 9
3<|x|<5
-5<x<-3 or 3<x<5
For the region -5<x<-3
|x+5| + |x-3| = 8
But for the region 3<x<5
|x+5| + |x-3| = 2 + 2x
NOT SUFFICIENT

IMO E

I tested a few numbers and got the answer right.
Although when I combined the two equations I interpreted it this way:
stat 1 says -5<x<5 and stat 2 says x>3 or x<-3

Thus, combining we can see from stat 1 that x<5 and from stat 2 that x<-3, thus combining these we see x<-3
Similarly combining the cases of what x should be bigger than, we get x>3
Thus, the final combined range becomes x>3 or x<-3 (essentially the same as stat 2), amd since stat 2 was anyway insuff, I marked E

Where did I go wrong in combining the two stats ? Experts pls help chetan2u VeritasKarishma Bunuel MathRevolution

Chitra657, that will not be the correct interpretation.

-5<x<5
…….-5|xxxxxxxxxxxxxx|5…..
x<-3 or x>3
xxxxxxxxx-3|………|3xxxxxxxxxx

x is the marked range of x
What is common in two that fits in both the ranges
…..-5|xxxxx|-3……..3|xxxxxxx|5…..

Do one thing - mark the inequalities you have obtained on the number line.
The blue line shows -5 < x < 5
The two black arrows show x > 3 or x < -3.

Which areas fall under both inequalities? (since both inequalities need to be satisfied)?
The 3 < x < 5 and -5 < x < -3

Hi there,
Out of interest, why would one assume that x is an integer when no where in the question does is state that x is an integer?
It seems like it is necessary to get to the answer in the right way, however I was under the impression that unless a question states that 'x is an integer', you should not assume it?
Thanks!

Bunuel

Could I please ask, how we can be sure that x = -4 here, as we are not given that x is an integer. Cant it be any value say -4.2, -3.5 etc?

For the range -5 ≤ x ≤ -3, the value of |x + 5| + |x - 3| is 8 regardless of the exact value of x. You can verify this by testing any x within -5 < x < -3. This happens because when -5 ≤ x ≤ 3, then x + 5 ≥ 0 and x - 3 ≤ 0, leading to |x + 5| = x + 5 and |x - 3| = -(x - 3). Therefore, in this range, |x + 5| + |x - 3| simplifies to x + 5 - (x - 3) = 8, which resolves to 8 = 8. Hence, |x + 5| + |x - 3|= 8 is valid for any value within this range.

Hope it's clear.

Bunuel

Can you please explain why you are checking if (x-3) is lesser than or greater than 9 and not 0 in each of the three ranges?

That was a typo. Thank you very much for noticing and reporting.