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Difficulty: HardSolving Linear Inequalities

A certain relationship between a number xx and other values is described as follows: one-fourth of the difference when 3x3x is subtracted from 22, decreased by one-third of the sum of xx and 33, is strictly greater than the difference when xx is subtracted from 11. Which of the following inequalities represents the complete set of all possible values of xx?

  1. A
    x<6x < 6
  2. B
    x>−18x > -18
  3. x<−18x < -18Answer
  4. D
    x>625x > \frac{6}{25}
  5. E
    x<−23x < -\frac{2}{3}

Answer

x<−18x < -18
To find the correct solution set, translate the word problem into the inequality 2−3x4−x+33>1−x\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x. First, multiply all terms by the least common denominator, 1212, to clear the fractions, giving 3(2−3x)−4(x+3)>12(1−x)3(2 - 3x) - 4(x + 3) > 12(1 - x). Distributing the constants yields 6−9x−4x−12>12−12x6 - 9x - 4x - 12 > 12 - 12x. Combining like terms on the left side simplifies the expression to −13x−6>12−12x-13x - 6 > 12 - 12x. Adding 12x12x to both sides results in −x−6>12-x - 6 > 12. Adding 66 to both sides gives −x>18-x > 18. Finally, dividing by −1-1 and reversing the inequality sign results in the solution set x<−18x < -18.

Step-by-Step Solution

1
Translate the verbal description into an algebraic inequality.
2−3x4−x+33>1−x\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x
To represent the relationships described in the word problem mathematically.
2
Multiply the entire inequality by the least common denominator, 12, to clear the fractions.
3(2−3x)−4(x+3)>12(1−x)3(2 - 3x) - 4(x + 3) > 12(1 - x)
Clearing denominators makes it easier to combine like terms and solve for xx.
3
Distribute the constants on both sides.
6−9x−4x−12>12−12x6 - 9x - 4x - 12 > 12 - 12x
To remove parentheses and separate individual terms for simplification.
4
Combine like terms on the left side of the inequality.
−13x−6>12−12x-13x - 6 > 12 - 12x
Simplifying the expressions on each side makes the inequality easier to solve.
5
Add 12x12x and 66 to both sides of the inequality to isolate variables on one side.
−x>18-x > 18
To group variable terms on the left and constant terms on the right.
6
Divide both sides by −1-1 and reverse the inequality sign.
x<−18x < -18
Dividing or multiplying an inequality by a negative number requires reversing the inequality sign to maintain a true statement.

Key Concept

Solving multi-step linear inequalities with rational terms and variable terms on both sides, including reversing the inequality sign when dividing by a negative number.

Alternative Method

Instead of clearing fractions first, you can distribute the division to each term (e.g., 24−3x4−x3−33>1−x\frac{2}{4} - \frac{3x}{4} - \frac{x}{3} - \frac{3}{3} > 1 - x), group the xx terms on one side and constants on the other using fraction arithmetic, and then solve for xx. However, clearing fractions with the LCD is generally faster and less prone to arithmetic errors.
Estimated Time:2m 0s
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Topics

Solving Linear Inequalities

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