Let S={(x, y): (y(3 x-1))/(x(3 x-2))<0} S^'={(x, y) ∈A ×B:-1...

Let $S=\left\{(x, y): \frac{y(3 x-1)}{x(3 x-2)}<0\right\}$
$S^{\prime}=\{(x, y) \in A \times B:-1 \leq A \leq 1,-1 \leq B \leq 1\}$
then the area of the region enclosed by all points in $S \cap S^{\prime}$ is
(a) 1
(b) 2
(c) 3
(d) 4

Key Concepts

Area Calculation for Composite Regions

Once the intersection of the defined regions is established, calculating its area involves either geometric reasoning or integration techniques. This concept emphasizes the ability to compute areas of regions that may be irregular or defined by complex boundaries, by either subdividing the region into simpler parts or by applying appropriate area formulas.

Intersection of Regions

This concept deals with finding the common set of points that simultaneously satisfy two or more conditions or constraints. In this problem, one set is defined by a rational inequality and the other is a bounded geometric shape, specifically a square. Understanding how to determine the overlap between these sets is key, as the intersection represents the region where both conditions hold.

Rational Inequalities

This concept involves analyzing and solving inequalities where the expression is a ratio of polynomials. It requires identifying when the numerator and the denominator individually are positive, negative, or zero, and then determining the overall sign of the rational expression across different intervals. The process often includes finding critical points that delineate where the expression changes sign.

Critical Point and Sign Chart Analysis

This technique is essential for solving rational inequalities. It involves finding the zeros of both the numerator and the denominator—points where the expression is zero or undefined—and then using these critical points to partition the number line into intervals. A sign chart is then constructed to assess the sign of the expression in each interval, which is critical for identifying the regions that satisfy the inequality.

Please give Ace some feedback