The graph of an inequality in the xy-plane consists of a... | SAT Linear inequalities in one or two variables Question, Answer Explained

Question bank / Linear inequalities in one or two variables

Math · AlgebraHard

The graph of an inequality in the $xy$ -plane consists of a solid line through the points $(0, 6)$ and $(4, 0)$ and the entire region below that line. Which inequality does the graph represent?

$2x + 3y \le 12$

$3x + 2y \le 12$

$3x + 2y \ge 12$

$3x - 2y \le 12$

Show answer and explanation

Answer: B

The line through $(0, 6)$ and $(4, 0)$ has slope $\frac{0 - 6}{4 - 0} = -\frac{3}{2}$ and $y$ -intercept $6$ , so it is $y = -\frac{3}{2}x + 6$ , or $3x + 2y = 12$ . A solid line with the region below it means $y \le -\frac{3}{2}x + 6$ , which is $3x + 2y \le 12$ . Check with $(0, 0)$ , which lies below the line: $0 \le 12$ is true.

Choice A switches the coefficients, which gives a line through $(0, 4)$ and $(6, 0)$ .

Why the other choices are wrong

A. This line crosses the axes at (0, 4) and (6, 0), not at (0, 6) and (4, 0).
C. This describes the region above the line. The point (0, 0), which is below the line, doesn't satisfy it.
D. This line has a positive slope, 3/2, but the line through (0, 6) and (4, 0) slopes downward.

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