To solve the inequality [tex]\((4x - 3)(2x - 1) \geq 0\)[/tex], you can follow these steps:
1. Identify the critical points:
The critical points occur when each factor equals zero. That is:
[tex]\[
4x - 3 = 0 \quad \text{and} \quad 2x - 1 = 0.
\][/tex]
Solving these equations, we get:
[tex]\[
x = \frac{3}{4} \quad \text{and} \quad x = \frac{1}{2}.
\][/tex]
2. Determine the intervals defined by the critical points:
The critical points divide the number line into three intervals:
[tex]\[
(-\infty, \frac{1}{2}), \quad (\frac{1}{2}, \frac{3}{4}), \quad \text{and} \quad (\frac{3}{4}, \infty).
\][/tex]
3. Test a point from each interval to determine the sign of the expression in that interval:
- For [tex]\( x < \frac{1}{2} \)[/tex], choose [tex]\( x = 0 \)[/tex]:
[tex]\[
(4 \cdot 0 - 3)(2 \cdot 0 - 1) = (-3)(-1) = 3 \geq 0.
\][/tex]
- For [tex]\( \frac{1}{2} < x < \frac{3}{4} \)[/tex], choose [tex]\( x = \frac{2}{3} \)[/tex]:
[tex]\[
\left( 4 \cdot \frac{2}{3} - 3 \right)\left( 2 \cdot \frac{2}{3} - 1 \right) = \left( \frac{8}{3} - 3 \right)\left( \frac{4}{3} - 1 \right) = \left( \frac{8}{3} - \frac{9}{3} \right)\left( \frac{4}{3} - \frac{3}{3} \right) = \left( -\frac{1}{3} \right)\left( \frac{1}{3} \right) = -\frac{1}{9} < 0.
\][/tex]
- For [tex]\( x > \frac{3}{4} \)[/tex], choose [tex]\( x = 1 \)[/tex]:
[tex]\[
(4 \cdot 1 - 3)(2 \cdot 1 - 1) = (4 - 3)(2 - 1) = 1 \geq 0.
\][/tex]
4. Combine the intervals where the expression is non-negative:
- From the tests above, the intervals that satisfy the inequality [tex]\((4x - 3)(2x - 1) \geq 0\)[/tex] are:
[tex]\[
(-\infty, \frac{1}{2}] \quad \text{and} \quad [\frac{3}{4}, \infty).
\][/tex]
Thus, the solution set to the inequality [tex]\((4x-3)(2x-1) \geq 0\)[/tex] is:
[tex]\[
\left\{ x \left| x \leq \frac{1}{2} \right. \text{ or } x \geq \frac{3}{4} \right\}.
\][/tex]
Therefore, the correct answer is:
[tex]\[
\left\{ x \left| x \leq \frac{1}{2} \right. \text{ or } x \geq \frac{3}{4} \right\}.
\][/tex]
