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Difficulty: MediumSolving Linear Inequalities

Which inequality represents all real values of xx for which the inequality 5−2x3≥x−62\frac{5 - 2x}{3} \geq \frac{x - 6}{2} is true?

  1. A
    x≥4x \geq 4
  2. B
    x≤113x \leq \frac{11}{3}
  3. x≤4x \leq 4Answer
  4. D
    x≥113x \geq \frac{11}{3}
  5. E
    x≤−87x \leq -\frac{8}{7}

Answer

The correct inequality is x≤4x \leq 4.
The correct inequality is x≤4x \leq 4. Multiplying both sides by the least common multiple, 66, yields the inequality 2(5−2x)≥3(x−6)2(5 - 2x) \geq 3(x - 6). Distributing the coefficients results in 10−4x≥3x−1810 - 4x \geq 3x - 18. Gathering the variable terms by subtracting 3x3x gives 10−7x≥−1810 - 7x \geq -18. Subtracting 1010 from both sides results in −7x≥−28-7x \geq -28. Dividing both sides by −7-7 and reversing the inequality sign results in the final solution x≤4x \leq 4.

Step-by-Step Solution

1
Multiply both sides of the inequality by 66 (the least common multiple of 22 and 33) to eliminate the fractions.
2(5−2x)≥3(x−6)2(5 - 2x) \geq 3(x - 6)
Eliminating denominators simplifies the linear inequality for solving.
2
Distribute the constants on both sides.
10−4x≥3x−1810 - 4x \geq 3x - 18
Expanding the terms allows combining like terms next.
3
Subtract 3x3x from both sides of the inequality.
10−7x≥−1810 - 7x \geq -18
Grouping all variable terms on one side of the inequality.
4
Subtract 1010 from both sides of the inequality.
−7x≥−28-7x \geq -28
Isolating the variable term on the left side.
5
Divide both sides by −7-7 and reverse the inequality sign.
x≤4x \leq 4
Dividing or multiplying both sides of an inequality by a negative number requires reversing the inequality sign direction.

Key Concept

Solving linear inequalities by clearing denominators and applying the sign-reversal rule when dividing by a negative number.
Estimated Time:1m 15s
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Topics

Solving Linear Inequalities

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