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Difficulty: MediumSolving Linear Inequalities

For all real values of xx, which of the following inequalities represents the complete solution set to 7−3(x−4)≥5x+317 - 3(x - 4) \ge 5x + 31?

  1. A
    x≥−32x \ge -\frac{3}{2}
  2. B
    x≤−92x \le -\frac{9}{2}
  3. x≤−32x \le -\frac{3}{2}Answer
  4. D
    x≥−92x \ge -\frac{9}{2}
  5. E
    x≤−254x \le -\frac{25}{4}

Answer

The inequality that represents the complete solution set is x≤−32x \le -\frac{3}{2}
The correct answer is the inequality stating that xx is less than or equal to negative three-halves. Simplifying the expression yields −8x≥12-8x \ge 12, and dividing by −8-8 correctly reverses the inequality direction to produce x≤−32x \le -\frac{3}{2}.

Step-by-Step Solution

1
Distribute −3-3 to both terms inside the parentheses on the left side of the inequality.
7−3x+12≥5x+317 - 3x + 12 \ge 5x + 31
To eliminate the parentheses and prepare the inequality for simplification.
2
Combine the constant terms 77 and 1212 on the left side.
19−3x≥5x+3119 - 3x \ge 5x + 31
To group like terms on the left side.
3
Subtract 5x5x from both sides of the inequality to collect variable terms on one side.
19−8x≥3119 - 8x \ge 31
To isolate the variable term on the left side.
4
Subtract 1919 from both sides of the inequality.
−8x≥12-8x \ge 12
To isolate the variable term completely before division.
5
Divide both sides of the inequality by −8-8 and flip the inequality sign.
x≤−128x \le -\frac{12}{8}, which simplifies to x≤−32x \le -\frac{3}{2}
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving multi-step linear inequalities, specifically reversing the inequality sign when multiplying or dividing by a negative number.
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Topics

Solving Linear Inequalities

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