Answer :
To solve the inequality [tex]\(\frac{1}{2x - 1} < 0\)[/tex], we need to determine when the expression is negative.
1. Understand the expression: [tex]\(\frac{1}{2x - 1}\)[/tex] is a fraction. A fraction is negative when its numerator and denominator have opposite signs. Since the numerator is a positive constant (1), we just need the denominator to be negative for the overall fraction to be negative.
2. Set up the inequality: To find when the denominator is negative, we set up the inequality:
[tex]\[
2x - 1 < 0
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Add 1 to both sides:
[tex]\[
2x < 1
\][/tex]
- Divide by 2:
[tex]\[
x < \frac{1}{2}
\][/tex]
4. Interval consideration: The inequality [tex]\(x < \frac{1}{2}\)[/tex] implies that [tex]\(x\)[/tex] can take any value less than [tex]\(\frac{1}{2}\)[/tex].
5. Combine with domain restrictions: There aren't explicit domain restrictions in this problem, but usually, we consider a reasonable interval for [tex]\(x\)[/tex]. The given options suggest we need to restrict [tex]\(x\)[/tex] between certain values, despite no specific lower boundary being required by the problem beyond typical real numbers.
6. Select the correct answer: According to the given choices, the interval consistent with [tex]\(x < \frac{1/2}\)[/tex] (without any additional domain coming from the question itself) is:
- The interval that fits this condition and is presented in the options is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
Therefore, the correct solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
1. Understand the expression: [tex]\(\frac{1}{2x - 1}\)[/tex] is a fraction. A fraction is negative when its numerator and denominator have opposite signs. Since the numerator is a positive constant (1), we just need the denominator to be negative for the overall fraction to be negative.
2. Set up the inequality: To find when the denominator is negative, we set up the inequality:
[tex]\[
2x - 1 < 0
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Add 1 to both sides:
[tex]\[
2x < 1
\][/tex]
- Divide by 2:
[tex]\[
x < \frac{1}{2}
\][/tex]
4. Interval consideration: The inequality [tex]\(x < \frac{1}{2}\)[/tex] implies that [tex]\(x\)[/tex] can take any value less than [tex]\(\frac{1}{2}\)[/tex].
5. Combine with domain restrictions: There aren't explicit domain restrictions in this problem, but usually, we consider a reasonable interval for [tex]\(x\)[/tex]. The given options suggest we need to restrict [tex]\(x\)[/tex] between certain values, despite no specific lower boundary being required by the problem beyond typical real numbers.
6. Select the correct answer: According to the given choices, the interval consistent with [tex]\(x < \frac{1/2}\)[/tex] (without any additional domain coming from the question itself) is:
- The interval that fits this condition and is presented in the options is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
Therefore, the correct solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
