A student translates a word problem into an inequality. The problem states: "One-fourth of… — ACT

A student translates a word problem into an inequality. The problem states: "One-fourth of the difference of $x$ and $3$ is greater than the sum of $1$ and one-third of the quantity $2x$ minus $1$ ." Which of the following inequalities represents the correct set of all real values of $x$ that satisfy this condition?

A
$x > -\frac{17}{5}$
B
$x < -\frac{44}{5}$
C
$x < -3$
$x < -\frac{17}{5}$ Answer
E
$x < -\frac{8}{3}$

Answer

The set of all real values of

x

satisfying the condition is

x < -\frac{17}{5}

The correct inequality is obtained by translating the word problem statement as

\frac{x - 3}{4} > 1 + \frac{2x - 1}{3}

. Multiplying the entire inequality by

12

yields

3(x - 3) > 12 + 4(2x - 1)

. Distributing the coefficients results in

3x - 9 > 12 + 8x - 4

, which simplifies to

3x - 9 > 8x + 8

. Moving all terms containing

x

to the left and constants to the right gives

-5x > 17

. Dividing both sides by

-5

and reversing the inequality sign results in the solution

x < -\frac{17}{5}

Step-by-Step Solution

Translate the verbal description into an algebraic inequality.

\frac{x - 3}{4} > 1 + \frac{2x - 1}{3}

The phrase "one-fourth of the difference of

x

and

3

" translates to

\frac{x - 3}{4}

, and "the sum of

1

and one-third of the quantity

2x

minus

1

" translates to

1 + \frac{2x - 1}{3}

Multiply all terms by the least common multiple of the denominators, which is

12

, to clear the fractions.

3(x - 3) > 12 + 4(2x - 1)

Multiplying both sides of an inequality by a positive number maintains the direction of the inequality sign while eliminating fractional coefficients.

Expand both sides of the inequality by distributing the coefficients.

3x - 9 > 12 + 8x - 4

Distribution allows the variable and constant terms to be separated and simplified.

Simplify the constants on the right side and move all variable terms to one side and constants to the other.

-5x > 17

Subtracting

8x

from both sides and adding

9

to both sides isolates the variable term on the left side of the inequality.

Divide both sides by

-5

and reverse the inequality sign.

x < -\frac{17}{5}

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating word problems and applying the sign-flip rule when dividing by a negative number.

Estimated Time:1m 30s

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