Question

graph the solution set of the compound inequality $4(1 - x) < -2$ and $\frac{x - 8}{3} leq -4$. choose the correct graph below.\
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\bigcirc a. \hspace{10em} \bigcirc b. \
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\bigcirc c. \hspace{10em} \bigcirc d. \
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select the correct choice below and, if necessary, fill in the answer box to complete your choice.\
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\bigcirc a. the solution set in interval notation is \square.\
\bigcirc b. the solution is the empty set.

Explanation:

Step1: Solve the first inequality \(4(1 - x) < -2\)

First, distribute the 4: \(4 - 4x < -2\). Then, subtract 4 from both sides: \(-4x < -2 - 4\), which simplifies to \(-4x < -6\). Divide both sides by -4 (remember to reverse the inequality sign when dividing by a negative number): \(x > \frac{-6}{-4} = \frac{3}{2} = 1.5\).

Step2: Solve the second inequality \(\frac{x - 8}{3} \leq -4\)

Multiply both sides by 3: \(x - 8 \leq -12\). Then, add 8 to both sides: \(x \leq -12 + 8 = -4\).

Step3: Analyze the solution sets

The first inequality gives \(x > 1.5\) and the second gives \(x \leq -4\). There is no number that is both greater than 1.5 and less than or equal to -4, so the solution set is the empty set.

Answer:

B. The solution is the empty set.