Solve for x: |3x-2|≤ ½  GMAT Problem Solving

Question: Solve for x: |3x-2|≤ ½

1. (½ , ⅚) 
2. (3/2 , 6/5) 
3. (5/2, ¾) 
4. (3/7 , 4/9) 
5. (6/7 , 4/3) 

Correct Answer: A

Solution and Explanation:

Approach Solution 1:

The problem statement asks to solve the equation |3x-2|≤ ½ for x. There are two cases that can be considered for the same.

To get the appropriate answer, the following are the two cases that have been assessed to get the correct answer. 

The first case has been evaluated as followed: 

The first case states that whatever is inside the modulus is >= 0 

⇒3x-2 ≥ 0 => 3x ≥ 2 => x ≥ ⅔ ~ 0.67
⇒|3x-2| = 3x-2 [with (as |X| = X when X >= 0)]
⇒ 3x-2 ≤ ½
⇒3x ≤ 2 + ½
⇒3x ≤ \(\frac{4+1}{2}\)
⇒ x ≤ \(\frac{5}{2*3}\)
⇒x ≤ 5/6

However, the given condition was that x \(\geq\) 2/3. 

Depending on this condition and the value of x determined from the above equation in case 1, it can be stated that- 

\(\frac{2}{3}\leq x \leq \frac{5}{6}\)

It is clear from the above equation that the value for x  is ⅔ and ⅚ 

However, another case that can be used to find the value for the x is as follows:

The second case states that whatever is inside the modulum < 0 

This can be further explained using the following equation and solving it- 

⇒3x-2 < 0 => 3x < 2 => x < \(\frac{2}{3}\)
⇒|3x-2| = -(3x-2) [ where |x| = -x and x < 0
⇒-(3x-2) ≤ \(\frac{1}{2}\)
⇒-3x + 2 ≤ \(\frac{1}{2}\)
⇒-3x ≤ \(\frac{1}{2}\) - 2
⇒-3x ≤ \(\frac{1-4}{2}\)
⇒-3x ≤  \(\frac{-3}{2}\)

⇒x ≥ \(\frac{-3}{2x-3}\) ( multiplying both the sides with -3 will help in reversing the sign of inequality because of which this situation has been considered) 
⇒x ≥ 2

But the given condition stated that x <\(\frac{2}{3}\) and x is not < \(\frac{2}{3}\)

Thus, in this case, no solution could be evaluated thereby stating that this case cannot be considered viable for solving the equation. 

Thus, considering both cases, it can be stated that Option A is the correct answer implying that the value of x is ⅔ and ⅚ which satisfies the first condition. 

Approach Solution 2:

The problem statement asks to solve the equation |3x-2|≤ ½ for x. 

Another approach to be considered for this problem is in the terms of both positive and negative evaluations.

This implies the following method to be employed- 

To derive the value of x, the opening of the mode with a positive sign would imply- 

⇒3x-2 ≤ 1/2
⇒x ≤ 5/6

This can be considered to be the value for x which is incorporated in one of the cases. 

Another case can be derived with the negative sign for opening the mode. This would imply- 

⇒-3x + 2 ≤ ½
⇒1/2 ≤ x

Hence, from both cases, the value of x is derived as ½ and ⅚. 

Approach Solution 3:

The problem statement asks to solve the equation |3x-2|≤ ½ for x. 

∣3x−2∣ ≤ 1​/2
3x−2 ≤1​/2
or 3x−2 ≥ −1​/2

Again, 3x−2 ≤ 1/2
6x−4 ≤ 1
6x ≤ 5
x ≤ 5/6

x ≤ 5​/6 or x ≥ 1​/2

Therefore, 1/2​ ≤  x ≤ 5​/6

Hence, the value of x is derived as ½ and ⅚

“Solve for x: |3x-2|≤ ½”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT Problem Solving questions are designed to test the calculative skills and intellectual level of the candidates to solve quantitative problems. GMAT Quant practice papers offer various types of questions that enable the candidates to strengthen their mathematical knowledge.

Suggested GMAT Problem Solving Questions

• A Watch Which Gains 5 Seconds In 3 Minutes Was Set Right At 7 A.M. GMAT Problem Solving
• Of The 800 Employees Of Company X, 70 Percent Have Been With The Company GMAT Problem Solving
• To 100 Litres Of Milk, 10 Litres Of Water Is Added And Then 20 Litres GMAT Problem Solving
• Walking at 3/4 of his normal speed, Mike is 16 minutes late GMAT Problem Solving
• There are 12 yes or no questions. How many ways can these be answered? GMAT Problem Solving
• There are 7 Red and 5 Blue Marbles in a Jar GMAT Problem Solving
• A Basket Contains 3 White and 5 Blue Balls GMAT Problem Solving
• Working together, John and Jack can type 20 pages in one hour GMAT Problem Solving
• Is x^2 *x^5*z>0? GMAT Problem Solving
• The average (arithmetic mean) of four distinct positive integers is 10 GMAT Problem Solving
• How many roots does the equation √x2+1+√x2+2=2 have? GMAT Problem Solving
• A Box Contains 10 Tablets of Medicine A and 15 Tablets of Medicine B GMAT Problem Solving
• If x=√10+3√9+4√8+5√7+6√6+7√5+8√4+9√3+10√2, then which of the following must be true? GMAT Problem Solving
• What is the last digit of (3)^(3)^3? GMAT Problem Solving
• It takes Jack 2 more hours than Tom to type 20 pages GMAT Problem Solving
• The angles in a triangle are x, 3x, 5x degrees GMAT Problem Solving
• If among 5 children, there are 2 siblings; in how many ways can the children be seated GMAT Problem Solving
• Among 200 People, 56% like Strawberry Jam, 44% like Apple Jam, and 40% like Raspberry Jam GMAT Problem Solving
• If 4 Women and 6 Men Work in the Accounting Department GMAT Problem Solving
• If 2^98=256L+N, where L and N are integers and 0≤N≤4, what is the value of N? GMAT Problem Solving
• A draining pipe can empty a pool in 4 hours GMAT Problem Solving
• If equation |x| + |y| = 5 encloses a certain region on the graph, what is the area of that region? GMAT Problem Solving
• Kate and David each have $10 GMAT Problem Solving
• How many five digit numbers can be formed using the digits 0, 1, 2, 3, 4, and 5 which are divisible by 3, without repeating the digits? GMAT Problem Solving
• Which of the following lines are parallel to line x = 4 – 2y? GMAT Problem Solving
• John has 12 clients and he wants to use color coding to identify each client 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
• If 69% of k is 23, approximately how much is 23% of k? GMAT Problem Solving
• Metropolis Corporation has 4 Shareholders GMAT Problem Solving
• If x and y are positive integers and (1/7)^x * (1/8)^12 = (1/8)^1/18y, then what is the value of y-x? GMAT Problem Solving
• At Springfield High, three-fourths of the male students and half of the female students speak a foreign language 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 GMAT Problem Solving
• A conference room is equipped with a total of 45 metal or wooden chairs GMAT Problem Solving
• A welder received an order to make a 1 million litre cube-shaped tank GMAT Problem Solving
• After 6 games, Team B had an average of 61.5 points per game GMAT Problem Solving
• If 12 ounces of a strong vinegar solution are diluted with 50 ounces of water to form a three-percent vinegar solution, what was the concentration of the original solution? 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
• A circle is inscribed in a square with the diagonal of 4 centimeters GMAT Problem Solving
• At a dog competition, a dog is awarded 10 points if it runs through 4 pipes, makes 10 jumps, and walks on 2 beams 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
• Simplify:\frac{4.5-2*\frac{3}{6}+\frac{1}{4^2}}{0.75} GMAT Problem Solving
• There are 6 points on xy-plane GMAT Problem Solving
• Mike, Tom, and Walt are working as sales agents for an insurance company GMAT Problem Solving
• A Commonwealth condominium complex has k apartments, n of which are rented at s dollars a month, providing total monthly revenue of p dollars. GMAT Problem Solving
• The Figure Below Shows a Square Inscribed in a Circle with the Radius of√6. What is the Area of the Shaded Region? GMAT Problem Solving