For what value of x is the equation x/5 - 4 = 3(4 - 2x) - 1 true?

For what value of x is the equation

2023 Handbook

2023 Test A Math · Question 44 of 57

Question

For what value of x is the equation $\frac{x}{5} - 4 = 3(4 - 2x) - 1$ true?

A $\frac{75}{11}$
B $\frac{75}{31}$
C $\frac{15}{7}$
D $\frac{65}{31}$

Show the answer

Correct answer: B

$\frac{x}{5} - 4 = 3(4 - 2x) - 1$	Apply the distributive property; multiply the 3 by 4 and $-2x$ .
$\frac{x}{5} - 4 = 12 - 6x - 1$	Combine like terms.
$\frac{x}{5} - 4 = 11 - 6x$	Apply the additive inverse property; add $6x$ and 4 to both sides of the equation.
$\frac{x}{5} + 6x = 15$	Get a common denominator.
$\frac{x + 30x}{5} = 15$	Apply the multiplicative inverse property; multiply both sides of the equation by 5.
$31x = 75$	Apply the multiplicative inverse property; divide both sides of the equation by 31.
$x = \frac{75}{31}$

Question analysis

This equation question asks for the value of x that makes x/5 − 4 = 3(4 − 2x) − 1 true. Students use the distributive property, collect variable terms on one side, and clear a fraction before solving. Each wrong choice matches a common slip with the parentheses or with the fraction, so careful bookkeeping matters more than speed.

NYS standards: 8.EE.7b linear equations with rational coefficients
7.EE.1 linear expressions with rational coefficients
Grade level: Grade 8 content, by NYS standard
Compared with grade 7: Above grade 7 Grade 7 equations have the form px + q = r. This one has x on both sides, a fractional coefficient and a product to expand, which are grade 8 equation skills, and every wrong choice matches a plausible slip along the way.
Estimated difficulty: Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Error-prone arithmetic moderate
Many steps moderate
A very close wrong answer minor
Solution: 5 steps
Reading level: Question and choices: about grade 8, 14 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every step is a routine move in solving a linear equation, but there are five of them and the fraction raises the cost of each. The listed wrong answers come from distributing the 3 to only one term, clearing the fraction from one term only, and stretching the parentheses over the −1. Substituting the result back is slow with a denominator of 31, so careful step-by-step work matters more than checking.

Error-prone arithmetic moderate Working it out
The term x/5 sits beside whole-number terms and a negative −6x, so clearing the fraction means multiplying every term by 5, and the answer is the unfriendly 75/31. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Many steps moderate Working it out
Five moves in a row, distributing the 3, combining 12 and −1, collecting the x terms, clearing the fifth and dividing by 31, each give a slip a place to enter. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
A very close wrong answer minor Choosing the answer
The fraction 75/11 shares its numerator with 75/31 and differs by one digit of the denominator, so only the coefficient of x, 6 1/5 rather than 2 1/5, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Distribute the 3 to both terms in the parentheses, combine constants, collect the x terms on one side, then multiply every term by 5 to clear the fraction; check by substituting.

Traps

A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

A A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: strong
Distributes the 3 only to the 4, writing 12 − 2x, so the equation becomes x/5 + 2x = 15 and then 11x = 75.
C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: moderate
Multiplies only x/5 by 5 to clear the fraction, leaving 6x and 15 unchanged, so it solves x + 6x = 15 and gets 7x = 15.
D Order of operations (Brown & Burton, 1978)
Reveals a misconception Expected pull: weak
Applies the 3 to the −1 as well, as if it sat inside the parentheses, so the right side becomes 9 − 6x and the constant 13 instead of 15.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

The distributive property: $a(b - c) = ab - ac$; a, b and c are numbers or terms. Multiplying a by a difference means multiplying a by each part. The number outside must reach every term inside the parentheses.
Doing the same to both sides: $\text{if } a = b\text{, then } a + c = b + c \text{ and } a \times c = b \times c$; a and b are the two sides of an equation, and c is any number. An equation is like a balanced scale. Adding the same amount to both sides, or multiplying both sides by the same number, keeps it balanced.

Sort out the question
You need the value of x that makes both sides of the equation equal. The plan: clear the parentheses, gather the x terms on one side, and get rid of the fraction. Then divide.
Clear the parentheses
$a(b - c) = ab - ac$
$3(4 - 2x) = 12 - 6x$
$\frac{x}{5} - 4 = 12 - 6x - 1 = 11 - 6x$
Watch out: the 3 multiplies both 4 and 2x. Writing 12 − 2x leads to $\frac{75}{11}$ , choice A. The −1 is outside the parentheses, so the 3 does not touch it.
Gather the x terms
Add 6x to both sides to move the x terms to the left. Add 4 to both sides to move the plain numbers to the right.
$\text{if } a = b\text{, then } a + c = b + c \text{ and } a \times c = b \times c$
$\frac{x}{5} + 6x = 11 + 4 = 15$
Clear the fraction
Multiply both sides by 5. Every term gets multiplied, not only the fraction.
$5 \times \frac{x}{5} + 5 \times 6x = 5 \times 15$
$x + 30x = 75$
$31x = 75$
Watch out: multiplying only the fraction by 5 gives x + 6x = 15. That leads to $\frac{15}{7}$ , choice C.
Divide by 31
$x = \frac{75}{31}$
75 and 31 share no factor except 1, so the fraction is already simplified. So the answer is B, $\frac{75}{31}$ .

Check your answer

Put $x = \frac{75}{31}$ back in. Then $\frac{x}{5} = \frac{15}{31}$ and $2x = \frac{150}{31}$ . Write 4 as $\frac{124}{31}$ . The left side is $\frac{15}{31} - \frac{124}{31} = -\frac{109}{31}$ . On the right, $4 - \frac{150}{31} = -\frac{26}{31}$ , and $3 \times \left(-\frac{26}{31}\right) - 1 = -\frac{78}{31} - \frac{31}{31} = -\frac{109}{31}$ . Both sides match.

The original DOE question

Question 101 as printed on page 86 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 86, question 101. Open page 86 of the handbook (PDF)