Graphical solution of linear inequalities in two variables

Correct answer: B. (2x+3y\ge18)

Writing the line as (y=\frac{18-2x}{3}), the upper side is (y\ge\frac{18-2x}{3}). This is equivalent to (2x+3y\ge18).

Correct answer: B. Bounded

The conditions (0\le x\le6) and (0\le y\le5) restrict the region inside a rectangle. The inequality (x+y\ge4) only cuts off one corner.

Correct answer: A. \( \left(\frac{13}{5},\frac{16}{5}\right) \)

Solving \(x+2y=9\) and \(3x+y=11\) gives \(x=\frac{13}{5}\), \(y=\frac{16}{5}\). Fractional vertices are common in graphical solutions.

Correct answer: A. A strip between two parallel lines

Both lines have slope (3), so they are parallel. The inequalities give a strip-like region between them.

Correct answer: A. (12) square units

The intercepts are ( (6,0) ) and ( (0,4) ), so the area is (\frac{1}{2}\times6\times4=12). For a triangular region, identify base and height.

Correct answer: C. (y\le0)

The (x)-axis is (y=0), and below it (y) is less than or equal to zero. Use (\le) when the boundary is included.

Correct answer: A. ( (1,2) ), ( (6,2) ), ( (1,7) )

The boundaries (x=1), (y=2), and (x+y=8) give ( (1,2) ), ( (6,2) ), and ( (1,7) ). Do not treat shifted restrictions as simple (x\ge0), (y\ge0).

Correct answer: C. Empty set

It is impossible for (x+y) to be both less than and greater than (5) at the same time. Opposite inequalities can make the solution empty.

Correct answer: D. ( (0,3) )

The vertices are ( (0,0) ), ( (3,0) ), ( (2,2) ), and ( (0,3) ), so every listed point is a vertex. The correct choice should have been none of these.

Correct answer: A. (y\ge\frac{x-9}{3})

From (x-3y\le9), we get (-3y\le9-x), hence (y\ge\frac{x-9}{3}). Dividing by a negative reverses the sign.

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

At ( (2,2) ), both left sides equal (10), which is less than (12). Therefore this option cannot be correct and true solutions must make both values at least (12).

Correct answer: A. \((2,4)\) and \((8,1)\)

On the boundary line \(x+2y=10\), put \(x=2\). Then \(2+2y=10\), so \(y=4\) and one vertex is \((2,4)\). Similarly, putting \(y=1\) gives \(x+2=10\), so \(x=8\) and the other vertex is \((8,1)\). The points in option D lie on the line, but they are not vertices formed by intersections with the boundary lines \(x=2\) and \(y=1\). Exam tip: find vertices by checking intersections of pairs of boundary lines.

Correct answer: A. \( \left(\frac{15}{4},\frac{15}{4}\right) \)

Adding the equations gives \(4x+4y=30\), and by symmetry \(x=y=\frac{15}{4}\). Use symmetry when boundary lines have a similar structure.

Correct answer: B. The solution region is empty

The value (x+y) cannot be at most (4) and at least (7) simultaneously. Parallel bounds in opposite positions can give an empty region.

Correct answer: C. ( (1,4) )

At ( (1,4) ), (1\le3) and (4\ge3) are true. Check both the vertical boundary and the oblique boundary for each option.

Correct answer: C. \( \left(\frac{14}{3},\frac{14}{3}\right) \)

The two equations give \(x=y\) and \(3x=14\), so the vertex is \( \left(\frac{14}{3},\frac{14}{3}\right) \). Symmetric boundaries can quickly give \(x=y\).

Correct answer: B. Below the line including the line

An inequality in the form (y\le) represents the region below the line. The equality part means the boundary line is included.

Correct answer: B. Unbounded region

Both inequalities select regions above the boundary lines, and the common region extends indefinitely. Conditions with (\ge) can often create an unbounded region.

Correct answer: B. (x\le4), (y\ge-1)

Left of (x=4) is (x\le4), and above (y=-1) is (y\ge-1). For vertical and horizontal boundaries, identify direction separately.

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

At ( (1,1) ), (x+y=2), which is less than (3). In multi-inequality questions, one false condition is enough to reject a point.

Correct answer: B. ( (4,1) )

At (y=1), (x\le8) and (x\le4), so the tighter limit is (x=4). For the right endpoint, choose the smaller upper bound from all constraints.

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

At ( (3,3) ), (6+9=15<18), so it is inside and not on the axes. For an interior point, look for a strict inequality value.

Correct answer: B. Rectangle

Two vertical and two horizontal boundaries together form a closed rectangle. In a graph, pairs of parallel boundaries help identify the shape quickly.

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

From (x+2=-x+4), (2x=2), so (x=1) and (y=3). The intersection of slant boundaries can mark the start or a corner of a region.

Correct answer: B. Toward origin with boundary

Putting the origin gives (0\leq 6), so the side containing it is taken. Exam tip: (\leq) includes the boundary line.