what is the solution set to the inequality $(4x-3)(2x-1) \ge 0$?

* $\{x \mid x \le 3 \text{ or } x \ge 1\}$
* $\{x \mid x \le 2 \text{ or } x \ge \frac{4}{3}\}$
* $\{x \mid x \le \frac{1}{2} \text{ or } x \ge \frac{3}{4}\}$
* $\{x \mid x \le -\frac{1}{2} \text{ or } x \ge -\frac{3}{4}\}$

Question

what is the solution set to the inequality $(4x-3)(2x-1) \ge 0$?

* $\{x \mid x \le 3 \text{ or } x \ge 1\}$
* $\{x \mid x \le 2 \text{ or } x \ge \frac{4}{3}\}$
* $\{x \mid x \le \frac{1}{2} \text{ or } x \ge \frac{3}{4}\}$
* $\{x \mid x \le -\frac{1}{2} \text{ or } x \ge -\frac{3}{4}\}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Inequalities" ], "new_concepts": [], "current_concepts": [ "Quadratic Inequalities" ] } ## Find the boundary points $$ \begin{aligned} (4x - 3)(2x - 1) &= 0 \ 4x - 3 = 0 &\implies x = \frac{3}{4} \ 2x - 1 = 0 &\implies x = \frac{1}{2} \end{aligned} $$ ## Determine the inequality intervals $$ \begin{aligned} &\text{For } (4x - 3)(2x - 1) \ge 0 \text{ with roots } x = \frac{1}{2} \text{ and } x = \frac{3}{4}: \ &x \le \frac{1}{2} \quad \text{or} \quad x \ge \frac{3}{4} \end{aligned} $$

(A) $\{x \mid x \le 3 \text{ or } x \ge 1\}$

(B) $\{x \mid x \le 2 \text{ or } x \ge \frac{4}{3}\}$

(C) $\{x \mid x \le \frac{1}{2} \text{ or } x \ge \frac{3}{4}\}$

(D) $\{x \mid x \le -\frac{1}{2} \text{ or } x \ge -\frac{3}{4}\}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Inequalities" ] }

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QUESTION IMAGE

Question

Explanation:

Find the boundary points

Determine the inequality intervals

Answer: