To determine which fractions are equivalent to [tex]\( -\frac{3}{2} \)[/tex], let's analyze each given fraction step by step:
1. [tex]\(\frac{3}{-2}\)[/tex]:
This fraction places the negative sign in the denominator. We can simplify it by recognizing that dividing by a negative number is equivalent to multiplying by a negative number in the numerator:
[tex]\[
\frac{3}{-2} = -\frac{3}{2}
\][/tex]
Thus, [tex]\(\frac{3}{-2}\)[/tex] is equivalent to [tex]\( -\frac{3}{2} \)[/tex].
2. [tex]\(-\left(\frac{-3}{2}\right)\)[/tex]:
In this fraction, the negative sign is outside the parenthesis and inside the parenthesis there is another negative fraction. Simplifying inside the parenthesis first:
[tex]\[
\frac{-3}{2} = -\frac{3}{2}
\][/tex]
Then applying the outer negative sign:
[tex]\[
-\left(-\frac{3}{2}\right) = \frac{3}{2}
\][/tex]
Hence, [tex]\(-\left(\frac{-3}{2}\right)\)[/tex] is not equivalent to [tex]\( -\frac{3}{2} \)[/tex].
3. [tex]\(\frac{-3}{-2}\)[/tex]:
This fraction has both the numerator and the denominator as negative numbers. A fraction with both parts negative can be simplified by changing the signs:
[tex]\[
\frac{-3}{-2} = \frac{3}{2}
\][/tex]
Thus, [tex]\(\frac{-3}{-2}\)[/tex] is not equivalent to [tex]\( -\frac{3}{2} \)[/tex].
4. [tex]\(-\left(\frac{3}{-2}\right)\)[/tex]:
In this fraction, the negative sign is outside the parenthesis, and inside the parenthesis is a fraction with a negative denominator:
[tex]\[
\frac{3}{-2} = -\frac{3}{2}
\][/tex]
Then applying the outer negative sign:
[tex]\[
-\left(-\frac{3}{2}\right) = \frac{3}{2}
\][/tex]
Thus, [tex]\(-\left(\frac{3}{-2}\right)\)[/tex] is not equivalent to [tex]\( -\frac{3}{2} \)[/tex].
In conclusion, the fraction that is equivalent to [tex]\( -\frac{3}{2} \)[/tex] is:
[tex]\[
\frac{3}{-2}
\][/tex]
