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Question 4 (Multiple Choice Worth 2 points)

Write and solve the inequality that represents [tex]\(-\frac{1}{4}\)[/tex] is less than the product of [tex]\(-\frac{2}{3}\)[/tex] and a number [tex]\(y\)[/tex].

A. [tex]\(-\frac{1}{4} \ \textless \ -\frac{2}{3} y\)[/tex] where [tex]\(y \ \textless \ \frac{3}{8}\)[/tex]
B. [tex]\(-\frac{1}{4} \leq -\frac{2}{3} y\)[/tex] where [tex]\(y \ \textless \ -\frac{2}{12}\)[/tex]
C. [tex]\(-\frac{2}{3} \ \textless \ -\frac{1}{4} y\)[/tex] where [tex]\(y \ \textless \ -\frac{3}{8}\)[/tex]
D. [tex]\(-\frac{2}{3} \ \textgreater \ -\frac{1}{4} y\)[/tex] where [tex]\(y \ \textgreater \ \frac{2}{12}\)[/tex]


Sagot :

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].