What Is The Solution Set To The Inequality $(4x - 3)(2x - 1) \geq 0$?A. $\{x \mid X \leq 3 \text{ Or } X \geq 1\}$B. $\{x \mid X \leq 2 \text{ Or } X \geq \frac{4}{3}\}$C. $\{x \mid X \leq \frac{1}{2} \text{ Or } X \geq
**What is the Solution Set to the Inequality (4x - 3)(2x - 1) ≥ 0?**
Understanding the Problem
The given inequality is a quadratic inequality in the form of (4x - 3)(2x - 1) ≥ 0. To find the solution set, we need to determine the values of x that satisfy this inequality.
Step 1: Find the Critical Points
To solve the inequality, we first need to find the critical points, which are the values of x that make the expression (4x - 3)(2x - 1) equal to zero. We can do this by setting each factor equal to zero and solving for x.
- 4x - 3 = 0 --> 4x = 3 --> x = 3/4
- 2x - 1 = 0 --> 2x = 1 --> x = 1/2
Step 2: Create a Sign Chart
Next, we create a sign chart to determine the intervals where the expression (4x - 3)(2x - 1) is positive or negative. We use the critical points x = 3/4 and x = 1/2 to divide the number line into three intervals: (-∞, 3/4), (3/4, 1/2), and (1/2, ∞).
| Interval | (4x - 3) | (2x - 1) | (4x - 3)(2x - 1) |
|---|---|---|---|
| (-∞, 3/4) | - | - | + |
| (3/4, 1/2) | + | - | - |
| (1/2, ∞) | + | + | + |
Step 3: Determine the Solution Set
Based on the sign chart, we can see that the expression (4x - 3)(2x - 1) is positive in the intervals (-∞, 3/4) and (1/2, ∞). Therefore, the solution set to the inequality (4x - 3)(2x - 1) ≥ 0 is x ≤ 3/4 or x ≥ 1/2.
Answer
The correct answer is A. {x | x ≤ 3/4 or x ≥ 1/2}.
Q&A
Q: What is the solution set to the inequality (4x - 3)(2x - 1) ≥ 0?
A: The solution set to the inequality (4x - 3)(2x - 1) ≥ 0 is x ≤ 3/4 or x ≥ 1/2.
Q: How do I find the critical points of the inequality?
A: To find the critical points, set each factor of the expression equal to zero and solve for x.
Q: What is a sign chart?
A: A sign chart is a table that shows the sign of the expression in different intervals.
Q: How do I determine the solution set using a sign chart?
A: Based on the sign chart, determine the intervals where the expression is positive or negative. The solution set is the union of these intervals.
Q: What is the difference between the solution set and the critical points?
A: The critical points are the values of x that make the expression equal to zero, while the solution set is the set of values of x that satisfy the inequality.
Q: Can I use a sign chart to solve any type of inequality?
A: Yes, a sign chart can be used to solve any type of inequality, including linear, quadratic, and rational inequalities.
Q: How do I know which intervals to use in the sign chart?
A: Use the critical points to divide the number line into intervals. Then, test a value from each interval in the expression to determine the sign of the expression in that interval.
Q: Can I use a sign chart to find the solution set to a system of inequalities?
A: Yes, a sign chart can be used to find the solution set to a system of inequalities. However, you will need to create a sign chart for each inequality and then find the intersection of the solution sets.
Q: How do I know if the solution set is a union or intersection of intervals?
A: If the inequality is "greater than or equal to" or "less than or equal to," the solution set is a union of intervals. If the inequality is "greater than" or "less than," the solution set is an intersection of intervals.