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Difficulty: Very hardSolving Linear Inequalities

What is the complete set of real numbers xx that satisfy the inequality 2−x3−3x−14≥2\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2?

  1. A
    x≥−1x \geq -1
  2. B
    x≤−1913x \leq -\frac{19}{13}
  3. C
    x≥54x \geq \frac{5}{4}
  4. x≤−1x \leq -1Answer
  5. E
    x≤913x \leq \frac{9}{13}

Answer

The complete set of real numbers satisfying the inequality is x≤−1x \leq -1.
To solve the inequality 2−x3−3x−14≥2\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2, we first eliminate the denominators by multiplying both sides by their least common multiple, which is 1212. This yields 4(2−x)−3(3x−1)≥244(2 - x) - 3(3x - 1) \geq 24. Distributing the terms gives 8−4x−9x+3≥248 - 4x - 9x + 3 \geq 24. Combining like terms results in 11−13x≥2411 - 13x \geq 24. Subtracting 1111 from both sides yields −13x≥13-13x \geq 13. Finally, dividing by −13-13 requires reversing the inequality sign, leading to the solution x≤−1x \leq -1.

Step-by-Step Solution

1
Multiply all terms on both sides of the inequality by 1212, which is the least common multiple of the denominators 33 and 44.
4(2−x)−3(3x−1)≥244(2 - x) - 3(3x - 1) \geq 24
Multiplying by a positive number preserves the inequality direction while eliminating the fractions to simplify the equation.
2
Distribute the coefficients 44 and −3-3 to the terms inside the parentheses.
8−4x−9x+3≥248 - 4x - 9x + 3 \geq 24
Applying the distributive property expands the expression. Note that multiplying −3-3 by −1-1 results in +3+3.
3
Combine like terms on the left side of the inequality.
11−13x≥2411 - 13x \geq 24
Grouping the constants (8+3=118 + 3 = 11) and the variable terms (−4x−9x=−13x-4x - 9x = -13x) simplifies the inequality.
4
Subtract 1111 from both sides of the inequality.
−13x≥13-13x \geq 13
Isolating the variable term on the left side by moving the constant term to the right side.
5
Divide both sides of the inequality by −13-13 and reverse the direction of the inequality sign.
x≤−1x \leq -1
Dividing by a negative number requires reversing the inequality sign from ≥\geq to ≤\leq.

Key Concept

Solving linear inequalities by clearing fractions, distributing terms correctly, and reversing the inequality sign when multiplying or dividing by a negative number.
Estimated Time:2m 0s
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Topics

Solving Linear Inequalities

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