Answer :
To solve the inequality [tex]\((4x - 3)(2x - 1) \geq 0\)[/tex], we need to find the intervals for [tex]\(x\)[/tex] where this inequality holds true. Here’s a step-by-step solution:
1. Find the critical points:
The inequality becomes equal to zero when either [tex]\(4x - 3 = 0\)[/tex] or [tex]\(2x - 1 = 0\)[/tex]. Solving these equations gives us the critical points.
- For [tex]\(4x - 3 = 0\)[/tex]:
[tex]\[ 4x = 3 \\ x = \frac{3}{4} \][/tex]
- For [tex]\(2x - 1 = 0\)[/tex]:
[tex]\[ 2x = 1 \\ x = \frac{1}{2} \][/tex]
2. Determine the sign of the expression in each interval:
These critical points divide the real number line into three intervals: [tex]\((-\infty, \frac{1}{2})\)[/tex], [tex]\((\frac{1}{2}, \frac{3}{4})\)[/tex], and [tex]\((\frac{3}{4}, \infty)\)[/tex].
3. Test each interval:
Pick a test point from each interval and determine the sign of [tex]\((4x - 3)(2x - 1)\)[/tex].
- Interval [tex]\((-\infty, \frac{1}{2})\)[/tex]:
Choose [tex]\(x = 0\)[/tex].
[tex]\[ (4(0) - 3)(2(0) - 1) = (-3)(-1) = 3 \quad (\text{positive}) \][/tex]
- Interval [tex]\((\frac{1}{2}, \frac{3}{4})\)[/tex]:
Choose [tex]\(x = \frac{2}{3}\)[/tex].
[tex]\[ \left(4\left(\frac{2}{3}\right) - 3\right)\left(2\left(\frac{2}{3}\right) - 1\right) = \left(\frac{8}{3} - 3\right)\left(\frac{4}{3} - 1\right) = \left(\frac{8}{3} - \frac{9}{3}\right)\left(\frac{4}{3} - \frac{3}{3}\right) = \left(-\frac{1}{3}\right)\left(\frac{1}{3}\right) = -\frac{1}{9} \quad (\text{negative}) \][/tex]
- Interval [tex]\((\frac{3}{4}, \infty)\)[/tex]:
Choose [tex]\(x = 1\)[/tex].
[tex]\[ (4(1) - 3)(2(1) - 1) = (4 - 3)(2 - 1) = 1 \times 1 = 1 \quad (\text{positive}) \][/tex]
4. Include the critical points:
Since the inequality is [tex]\(\geq 0\)[/tex], the critical points where the expression is zero ([tex]\(x = \frac{1}{2}\)[/tex] and [tex]\(x = \frac{3}{4}\)[/tex]) are included in the solution set.
Using this information, we can determine that the expression is non-negative in the intervals [tex]\(x \leq \frac{1}{2}\)[/tex] and [tex]\(x \geq \frac{3}{4}\)[/tex].
Therefore, the solution set for the inequality [tex]\((4x - 3)(2x - 1) \geq 0\)[/tex] is:
[tex]\[ \left\{ x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4} \right\} \][/tex]
1. Find the critical points:
The inequality becomes equal to zero when either [tex]\(4x - 3 = 0\)[/tex] or [tex]\(2x - 1 = 0\)[/tex]. Solving these equations gives us the critical points.
- For [tex]\(4x - 3 = 0\)[/tex]:
[tex]\[ 4x = 3 \\ x = \frac{3}{4} \][/tex]
- For [tex]\(2x - 1 = 0\)[/tex]:
[tex]\[ 2x = 1 \\ x = \frac{1}{2} \][/tex]
2. Determine the sign of the expression in each interval:
These critical points divide the real number line into three intervals: [tex]\((-\infty, \frac{1}{2})\)[/tex], [tex]\((\frac{1}{2}, \frac{3}{4})\)[/tex], and [tex]\((\frac{3}{4}, \infty)\)[/tex].
3. Test each interval:
Pick a test point from each interval and determine the sign of [tex]\((4x - 3)(2x - 1)\)[/tex].
- Interval [tex]\((-\infty, \frac{1}{2})\)[/tex]:
Choose [tex]\(x = 0\)[/tex].
[tex]\[ (4(0) - 3)(2(0) - 1) = (-3)(-1) = 3 \quad (\text{positive}) \][/tex]
- Interval [tex]\((\frac{1}{2}, \frac{3}{4})\)[/tex]:
Choose [tex]\(x = \frac{2}{3}\)[/tex].
[tex]\[ \left(4\left(\frac{2}{3}\right) - 3\right)\left(2\left(\frac{2}{3}\right) - 1\right) = \left(\frac{8}{3} - 3\right)\left(\frac{4}{3} - 1\right) = \left(\frac{8}{3} - \frac{9}{3}\right)\left(\frac{4}{3} - \frac{3}{3}\right) = \left(-\frac{1}{3}\right)\left(\frac{1}{3}\right) = -\frac{1}{9} \quad (\text{negative}) \][/tex]
- Interval [tex]\((\frac{3}{4}, \infty)\)[/tex]:
Choose [tex]\(x = 1\)[/tex].
[tex]\[ (4(1) - 3)(2(1) - 1) = (4 - 3)(2 - 1) = 1 \times 1 = 1 \quad (\text{positive}) \][/tex]
4. Include the critical points:
Since the inequality is [tex]\(\geq 0\)[/tex], the critical points where the expression is zero ([tex]\(x = \frac{1}{2}\)[/tex] and [tex]\(x = \frac{3}{4}\)[/tex]) are included in the solution set.
Using this information, we can determine that the expression is non-negative in the intervals [tex]\(x \leq \frac{1}{2}\)[/tex] and [tex]\(x \geq \frac{3}{4}\)[/tex].
Therefore, the solution set for the inequality [tex]\((4x - 3)(2x - 1) \geq 0\)[/tex] is:
[tex]\[ \left\{ x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4} \right\} \][/tex]
