To solve the inequality [tex]\( -\frac{1}{4} < -\frac{2}{3} y \)[/tex], we must isolate [tex]\( y \)[/tex].
Given the inequality [tex]\( -\frac{1}{4} < -\frac{2}{3} y \)[/tex]:
1. Divide both sides of the inequality by [tex]\( -\frac{2}{3} \)[/tex]:
[tex]\[
-\frac{1}{4} < -\frac{2}{3} y
\][/tex]
2. Since we are dividing by a negative number, we must reverse the inequality sign:
[tex]\[
y < \frac{-\frac{1}{4}}{-\frac{2}{3}}
\][/tex]
3. Simplify the right side of the inequality:
[tex]\[
y < \frac{-\frac{1}{4}}{-\frac{2}{3}} = \frac{1/4}{2/3}
\][/tex]
4. Division of fractions means multiplying by the reciprocal:
[tex]\[
y < \frac{1}{4} \times \frac{3}{2} = \frac{3}{8}
\][/tex]
So, the inequality simplifies to:
[tex]\[
y < \frac{3}{8}
\][/tex]
From the initial problem and the final multiple-choice answers, the correct representation and solution for the inequality are found. The inequality equivalent to our derived inequality but phrased differently matches:
[tex]\( -\frac{2}{3} > -\frac{1}{4} y \)[/tex]
When [tex]\( y \)[/tex] is replaced with our calculated and simplified value:
So:
[tex]\( -\frac{2}{3} > -\frac{1}{4} y \)[/tex] where [tex]\( y = \frac{2}{12} \)[/tex].
Thus, the correct answer is:
[tex]\( -\frac{2}{3} > -\frac{1}{4} y \)[/tex] where [tex]\( y > \frac{2}{12} \)[/tex].
