Graphical solution of linear inequalities in two variables

Correct answer: D. ((0,4))

At ((0,4)), (x-y=-4), which does not satisfy (x-y \ge 0). Check every possible vertex against all boundaries.

Correct answer: B. (y>2x-3)

The region above the line means larger (y), and the boundary is excluded, so (>) is used. Read direction and line type together.

Correct answer: C. \(\left(\frac{10}{3},\frac{10}{3}\right)\)

The slant lines intersect at \(\left(\frac{10}{3},\frac{10}{3}\right)\), where \(x+y=\frac{20}{3}\). Check linear expressions at vertices for maxima.

Correct answer: A. ((0,0))

Substituting ((0,0)) gives (0 \ge 6), which is false, so the origin side is wrong. Shade the side opposite to a false test point.

Correct answer: B. (x \ge 1), (x \le 5), (y \ge 2), (y \le 6)

Lower and upper bounds on both (x) and (y) close the region on all sides. Four closed boundaries give a rectangular quadrilateral.

Correct answer: C. This point is not in the solution region

At ((2,3)), (x+2y=8), but the inequality is (<8), so the boundary is excluded. Points on the boundary are not solutions for strict inequalities.

Correct answer: A. (y \ge x) and (y \le -x+6)

In the between-region, (y) is above (y=x) and below (y=-x+6). Identify the lower and upper boundaries in such cases.

Correct answer: A. ((2,7))

Putting (x=2) in (x+y=9) gives (y=7), so ((2,7)) is a vertex. Use pairs of boundary lines to find possible vertices.

Correct answer: B. Origin is on boundary and included in solution

Substituting ((0,0)) gives (0 \ge 0), true, and equality means the point is on the boundary. The boundary is included for (\ge).

Correct answer: C. (3x-y \ge 9)

A solid line requires (\le) or (\ge), and at ((0,0)), (0 \ge 9) is false. Hence the origin is not included.

Correct answer: B. (3)

The vertices are ((1,1)), ((5,1)), and ((1,5)), so there are (3). Count valid intersections of all boundary lines.

Correct answer: B. (4)

For maximum (y), put (x=0), then (3y=12) and (y=4). Axis intercepts help identify maximum limits.

Correct answer: C. ((2,3))

At ((2,3)), (x+2y=8), which is less than (10), so it lies in the lower half-plane. Below can also be checked by smaller (y)-values.

Correct answer: B. ((3,2))

Adding (x-y=1) and (2x+y=8) gives (3x=9), so (x=3) and (y=2). Convert boundaries to equalities and solve.

Correct answer: B. \((16,0)\)

The \(x\)-intercept is the point where the line meets the \(x\)-axis, so \(y=0\) there. Substituting \(y=0\) in \(x+4y=16\) gives \(x=16\). Hence, the \(x\)-intercept is \((16,0)\). The point \((0,4)\) is the \(y\)-intercept, not the \(x\)-intercept. Exam tip: to find an \(x\)-intercept, always set \(y=0\).

Correct answer: A. (x \le 3)

On the (x)-axis, (y=0), so (2x \le 6) and (x \le 3). If (x \ge 0) is not separately given, negative (x)-values may also be included.

Correct answer: A. Closed strip between two parallel lines

Both lines have the same slope and equality is included, so a closed strip between them is obtained. Parallel boundaries form a strip.

Correct answer: B. Cut part of an unbounded strip

The first two inequalities form a parallel strip, and the third cuts it from one side but does not close it fully. Hence the region remains unbounded.

Correct answer: C. ((3,1))

At ((3,1)), (1 \le 2) and (1 \ge -2), both true. A solution point must lie between the two boundaries.

Correct answer: C. (x-3y \ge 6)

Substituting ((0,-3)) gives (0-3(-3)=9), so (9 \ge 6) is true and (\ge) is correct for a solid line. Combine the test point with line type.

Correct answer: B. (4)

The vertices are ((0,0)), ((4,0)), \(\left(\frac{8}{3},\frac{8}{3}\right)\), and ((0,4)). List all valid corners in order.

Correct answer: B. Their common solution is empty

The same expression cannot be both greater than (5) and less than (5). Opposite strict inequalities make the common solution empty.

Correct answer: C. (4)

The boundary line is (2x+ay=12), and substituting ((0,3)) gives (3a=12), so (a=4). Put a boundary point in the equality.

Correct answer: A. \(\left(\frac{22}{5},\frac{24}{5}\right)\)

Solving \(x+2y=14\) and \(3x+y=18\) gives \(x=\frac{22}{5}\), \(y=\frac{24}{5}\). In a graph, always find the intersection of slant boundaries separately.

Correct answer: B. Point \(\left(\frac{8}{3},\frac{8}{3}\right)\)

Solving the two boundary lines together gives \(x=y=\frac{8}{3}\). In exams, find vertices by treating boundary lines as equations.