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Difficulty: HardSolving Linear Inequalities

For all real values of xx, which of the following inequalities represents the complete solution set to the inequality 5−2x3−3x−14≤x+22\frac{5 - 2x}{3} - \frac{3x - 1}{4} \leq \frac{x + 2}{2}?

  1. A
    x≤1123x \leq \frac{11}{23}
  2. B
    x≥523x \geq \frac{5}{23}
  3. C
    x≥−217x \geq -\frac{2}{17}
  4. x≥1123x \geq \frac{11}{23}Answer
  5. E
    x≤523x \leq \frac{5}{23}

Answer

The complete solution set is the set of all real numbers greater than or equal to 11/23.
The correct answer is found by clearing the denominators with the least common multiple of 12, carefully expanding the terms to get 20−8x−9x+3≤6x+1220 - 8x - 9x + 3 \leq 6x + 12, simplifying to 23−17x≤6x+1223 - 17x \leq 6x + 12, grouping terms to get −23x≤−11-23x \leq -11, and dividing by −23-23 which flips the sign to yield all real values greater than or equal to 11/23.

Step-by-Step Solution

1
Multiply all terms of the inequality by the least common multiple of the denominators (3, 4, and 2), which is 12, to clear the fractions.
4(5−2x)−3(3x−1)≤6(x+2)4(5 - 2x) - 3(3x - 1) \leq 6(x + 2)
Multiplying by a positive number allows us to eliminate denominators without changing the direction of the inequality.
2
Distribute the coefficients on both sides of the inequality, paying close attention to the distribution of the negative sign over the second term.
20−8x−9x+3≤6x+1220 - 8x - 9x + 3 \leq 6x + 12
Distributing −3-3 to both 3x3x and −1-1 yields −9x-9x and +3+3 respectively.
3
Combine the constant terms and the variable terms on the left side of the inequality.
23−17x≤6x+1223 - 17x \leq 6x + 12
Simplifying the expressions on each side makes the inequality easier to isolate.
4
Isolate the variable terms on the left and the constant terms on the right by subtracting 6x6x and 23 from both sides.
−23x≤−11-23x \leq -11
Grouping like terms together is necessary to solve for the variable.
5
Divide both sides of the inequality by −23-23 and reverse the direction of the inequality sign.
x≥1123x \geq \frac{11}{23}
Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign from ≤\leq to ≥\geq.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients, specifically applying the rule that multiplying or dividing by a negative number reverses the inequality direction.

Alternative Method

Instead of clearing the fractions first, write each fraction as separate terms: 53−23x−34x+14≤12x+1\frac{5}{3} - \frac{2}{3}x - \frac{3}{4}x + \frac{1}{4} \leq \frac{1}{2}x + 1. Then, collect the constant terms on one side and the variable terms on the other side using decimal or fractional conversions, and isolate the variable.
Estimated Time:2m 0s
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Topics

Solving Linear Inequalities

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