If x^2−5x+6=2−|x−1|, What is the Product of all Possible Values of x? GMAT Problem Solving

Question: If \(x^2-5x+6=2-|x-1|\), what is the product of all possible values of x?

1. 1
2. 3
3. 4
4. 5
5. 15

Answer:

Solution with Explanation:
Approach Solution (1):

One critical point at x = 1. So two ranges will be there

(1) x < 1

\(x^2-5x+6=2-|x-1|=x^2-5x+6=2--(x-1)\)
\(x^2-5x+6=2+x-1\)
\(\Rightarrow x^2-6x+5=0\)
OR
\(x^2-5x-x+5=0\)
OR
(x-5)(x-1)=0

But since we are looking at the range x < 1, both 1 and 5 are not valid.

(2) x >= 1

\(x^2-5x+6=2-|x-1|=x^2-5x+6=2-(x-1)\)
\(x^2-5x+6=2-x+1\)
\(\Rightarrow x^2-4x+3=0\)

OR

\(x^2-3x-x+3=0\)

OR

(x-3)(x-1)=0
x = 3 or x = 1

As we are looking at the range x >= , both 3 and 1 are valid solution
So product = 1 * 3 = 3

Correct Option: B

Approach Solution (2):

We have found the different values of x correctly as 5, 1, 3, 1…

But which are the valid values- you will have to substitute and check it.

Since you are dealing with modulus.

\(x^2-5x+6=2-|x-1|\)

(1) 5 –

\(5^2-5*5+6=2-|5-1|\)
25 – 25 + 6 = 2 – 4 or 6 = -4

No, so 5 is not valid.

(2) 1 –

\(1^2-5*1+6=2-|1-1|\)
2 = 2...

Valid

(3) 3 –

\(3^2-5*3+6=2-|3-1|\)
9 – 15 + 6 = 2 – 2

OR

0=0...

Valid

So there are two solutions 1 and 3.

Product = 1 * 3 = 3

Correct Option: B

Approach Solution (3):

We can solve it by isolating the modulus from the rest of the equation and solving two different quadratic equations:

\(x-1 = x^2-5x+4 \rightarrow x^2-6x+5=0 \rightarrow x_1 = 5, x_2 = 1\)
\(x-1 = -x^2-5x+4 \rightarrow -x^2-4x-3=0 \rightarrow x_1 = -1, x_2 = -3\)

Product = 3 * 5 = 15

Correct Option: E

“If, what is the product of all possible values of x?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Suggested GMAT Problem Solving Samples

• Is Square Root = Always Positive? GMAT Problem Solving
• \(\frac{15^{11}-15^{10}}{14}=?\) GMAT Problem Solving
• For How Many Values of k is 12^12 the Least Common Multiple GMAT Problem Solving
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving
• If n = \(3^8\)– \(2^8\), Which of the Following is Not a Factor of n? GMAT Problem Solving
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours. GMAT Problem Solving
• GMAT Problem Solving - The Price of Raw Materials Has Gone up by 15%
• Greg Assembles Units of a Certain Product at a Factory. GMAT Problem Solving
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving
• Find The Number Of Trailing Zeros In The Product Of (1^1) GMAT Problem Solving
• Find The Altitude Of An Equilateral Triangle Whose Side is 20 GMAT Problem Solving
• Which of the following is the value of √3√0.000064? GMAT Problem Solving
• Two Dice are Thrown Simultaneously. What is the Probability of Getting Two Numbers Whose Product is Even? GMAT Problem Solving
• The Smallest 3-Digit Positive Integer Obtained By Adding Two Positive Two-Digit Numbers GMAT Problem Solving
• Train A Leaves New York at 9am Eastern Time on Monday, Headed for Los Angeles at a Constant Rate GMAT Problem Solving
• An Express Train Traveled At An Average Speed of 100 Kilometers Per Hour GMAT Problem Solving
• If m Lies Between The Integers p and s On the Number Line Shown GMAT Problem Solving
• A Cube Of Side 7 cm Is Coloured On Pair of Opposite Faces By Red, Green and Yellow Shades GMAT Problem Solving
• If x^2 − 5x − 6 = 0, which of the following could be x ? GMAT Problem Solving
• How Many Numbers Between 1 and 1000, Inclusive Have an Odd Number of Factors? GMAT Problem Solving
• A palindrome is a number that reads the same forward and backward GMAT Problem Solving
• Three Cars Leave From A To B In Equal Time Intervals GMAT Problem Solving
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes GMAT Problem Solving
• A Zookeeper Counted the Heads of the Animals in a Zoo and Found it to be 80 GMAT Problem Solving
• A Shop Stores x kg of Rice. The First Customer Buys half this Amount Plus half a kg of Rice GMAT Problem Solving
• In a Class of 120 Students Numbered 1 to 120, All Even Numbered Students Opt for Physics GMAT Problem Solving
• Machine A Produces bolts at a Uniform Rate of 120 Every 40 seconds GMAT Problem Solving
• Out of 7 Consonants and 4 Vowels, How Many Words of 3 Consonants and 2 Vowels Can be Formed? GMAT Problem Solving
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 Seconds GMAT Problem Solving
• 12 Marbles are Selected at Random from a Large Collection of White, Red, Green and Yellow Marbles GMAT Problem Solving
• Find the greatest number that will divide 43, 91 and 183 GMAT Problem Solving
• Of the 150 Houses in a Certain Development GMAT Problem Solving
• A man can hit a target once in 4 shots. If he fires 4 shots in succession GMAT Problem Solving
• If 75 Percent of a Class Answered the First Question on a Certain Test Correctly GMAT Problem Solving
• A Clock Strikes 4 taking 9 seconds. GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• Find The Value Of x GMAT Problem Solving
• GMAT Problem Solving - Which of the following expressions has the greatest values?
• GMAT Problem Solving – If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)?
• GMAT Problem Solving – What is the product of all possible solutions of the equation |x+2|- 5|x+2| = -6?
• GMAT Problem Solving – Metropolis Corporation has 4 Shareholders
• GMAT Problem Solving – What is the number of integers from 1 to 1000, inclusive that are not divisible by 11 or by 35?
• GMAT Problem Solving – If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m?
• GMAT Problem Solving – If Ben Were to Lose the Championship, Mike would be the Winner
• GMAT Problem Solving – A Train Travelling at a Certain Constant Speed takes 30 seconds
• What is the Value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2 GMAT Problem Solving
• If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)? GMAT Problem Solving 
• Metropolis Corporation has 4 Shareholders GMAT Problem Solving 
• A Regular Hexagon has a Perimeter of 30 units GMAT Problem Solving
• All the Numbers 2, 3, 4, 5, 6, 7 are Assigned to the Six Faces of a Cube, One Number to Each Face GMAT Problem Solving