Final answer:
The solution to the inequality (4x-3)(2x-1) ≥ 0 is the set of all x values that make the expression non-negative, which is x ≤ 1/2 or x ≥ 3/4.
Step-by-step explanation:
The solution to the inequality (4x-3)(2x-1) ≥ 0 involves finding the values of x where the product of these two expressions is non-negative.
Firstly, determine the zeros of each expression by setting them equal to zero:
4x-3 = 0 → x = 3/4
2x-1 = 0 → x = 1/2
These are the critical points that divide the number line into intervals. Test each interval to see whether the inequality holds true:
- For x < 1/2, both factors (4x-3) and (2x-1) are negative, so their product is positive, satisfying the inequality.
- For 1/2 < x < 3/4, (2x-1) is positive and (4x-3) is negative, so their product is negative, not satisfying the inequality.
- For x > 3/4, both factors are positive, so their product is positive, satisfying the inequality.
Therefore, the solution set to the inequality is x ≤ 1/2 or x ≥ 3/4.