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Difficulty: MediumSolving Linear Inequalities

Which of the following inequalities represents the complete solution set for xx in the inequality x3−2x−52≥2\frac{x}{3} - \frac{2x - 5}{2} \ge 2?

  1. x≤34x \le \frac{3}{4}Answer
  2. B
    x≥34x \ge \frac{3}{4}
  3. C
    x≤3x \le 3
  4. D
    x≤−274x \le -\frac{27}{4}
  5. E
    x≥3x \ge 3

Answer

The inequality is satisfied by all values of xx less than or equal to 34\frac{3}{4}.
The correct solution is obtained by rewriting the fractions with a common denominator of 6, which yields 2x−3(2x−5)6≥2\frac{2x - 3(2x - 5)}{6} \ge 2. Distributing the negative sign gives −4x+156≥2\frac{-4x + 15}{6} \ge 2. Multiplying both sides by 6 and subtracting 15 results in −4x≥−3-4x \ge -3. Dividing by −4-4 and reversing the inequality sign yields x≤34x \le \frac{3}{4}.

Step-by-Step Solution

1
Find a common denominator of 6 for the two fractions on the left side of the inequality.
2x6−3(2x−5)6≥2\frac{2x}{6} - \frac{3(2x - 5)}{6} \ge 2
Finding a common denominator is necessary to combine fractional terms with different denominators.
2
Combine the fractions into a single expression, distributing the subtraction sign to both terms in the numerator of the second fraction.
2x−(6x−15)6≥2  ⟹  −4x+156≥2\frac{2x - (6x - 15)}{6} \ge 2 \implies \frac{-4x + 15}{6} \ge 2
Combining the fractions simplifies the inequality. Distributing the negative sign ensures correct operations on the terms.
3
Multiply both sides of the inequality by 6.
−4x+15≥12-4x + 15 \ge 12
Multiplying by a positive number isolates the numerator without changing the direction of the inequality sign.
4
Subtract 15 from both sides of the inequality.
−4x≥−3-4x \ge -3
This isolates the variable term on the left side.
5
Divide both sides by −4-4 and reverse the direction of the inequality sign.
x≤34x \le \frac{3}{4}
Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain the correct relationship.

Key Concept

Solving linear inequalities requires applying standard algebraic operations while remembering to reverse the inequality sign when multiplying or dividing both sides by a negative number.

Alternative Method

Instead of combining fractions first, multiply every term on both sides of the inequality by 6 (the least common multiple of the denominators) to eliminate the fractions immediately: 6⋅(x3)−6⋅(2x−52)≥6⋅26 \cdot \left(\frac{x}{3}\right) - 6 \cdot \left(\frac{2x - 5}{2}\right) \ge 6 \cdot 2. This simplifies directly to 2x−3(2x−5)≥122x - 3(2x - 5) \ge 12, which reduces to −4x+15≥12-4x + 15 \ge 12.
Estimated Time:1m 30s
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Topics

Solving Linear Inequalities

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