Graphical solution of linear inequalities in two variables

Correct answer: A. Solid line and half-plane containing origin

The sign (\le) includes the boundary line and substituting ((0,0)) gives (0 \le 12). In exams, test the origin first.

Correct answer: B. Dotted line on the side containing ((5,0))

The sign (>) excludes the boundary line and substituting ((5,0)) gives (5>4). Use a dotted line for strict inequalities.

Correct answer: A. ((0,0)), ((7,0)), ((0,7))

In the first quadrant, (x+y=7) cuts the axes at ((7,0)) and ((0,7)). Intersections of axes and the line give the vertices.

Correct answer: C. ((5,2))

The direct answer is option C: (5,2) lies in the half-plane 3x + 2y ≥ 18. Test each point by substitution. For A, (2,3): 3(2) + 2(3) = 6 + 6 = 12, and 12 ≥ 18 is false. For B, (4,2): 12 + 4 = 16, so it is false. For C, (5,2): 15 + 4 = 19, and 19 ≥ 18 is true. For D, (1,5): 3 + 10 = 13, so it is false. Therefore only option C belongs to the required half-plane. The equality line 3x + 2y = 18 is the boundary; points on the side giving a value greater than 18 satisfy the inequality, and points giving a smaller value do not. Exam cue: for a proposed point, substitute first and compare directly.

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

Equating boundary lines gives (2x+1=-x+4), so (x=1) and (y=3). A corner on the graph often comes from solving boundary lines.

Correct answer: A. (y>4x-8), above the line

From (4x-y<8), (-y<8-4x), so reversing the sign gives (y>4x-8). When dividing by a negative number, the inequality sign reverses.

Correct answer: B. (2x-y < 5)

A dotted line indicates a strict inequality and substituting ((0,0)) gives (0<5). The test point in the shaded side decides the correct sign.

Correct answer: D. ((3,4))

At ((3,4)), (x+2y=11), so (11 \le 10) is false. A point must satisfy every inequality in the system.

Correct answer: B. Above horizontal line and left of vertical line

(y \ge 3) gives the region above (y=3) and (x \le 5) gives the region left of (x=5). Read vertical and horizontal boundaries separately.

Correct answer: B. ((4,2))

Subtracting (x+y=6) from (2x+y=10) gives (x=4), then (y=2). Corners of a common graph come from boundary lines.

Correct answer: C. ((4,3))

At ((4,3)), (x-y=1) and (x+y=7), so both conditions are true. Algebraic checking is faster before graphing.

Correct answer: A. ((4,0)) and ((0,10))

Putting (y=0) gives (x=4), and putting (x=0) gives (y=10). Intercepts are the easiest way to draw the boundary line.

Correct answer: B. (x+3y \le 9)

Below the line means (y \le 3-\frac{x}{3}), and including the boundary requires (\le). A solid line is used for included boundaries.

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

At ((3,2)), (2x+y=8) and (x+y=5), so all conditions are satisfied. In a common solution, every inequality must be true.

Correct answer: A. Triangle

The three boundary lines give vertices ((1,2)), ((6,2)), and ((1,7)). A closed region from three lines is generally a triangle.

Correct answer: B. Boundary line is dotted and region is below

The strict sign (<) makes the line dotted, and smaller (y) means the region below the line. Use the sign to decide both line type and direction.

Correct answer: B. Side opposite to origin

If the test point gives false, the shading is on the other side of the boundary line. Even a false test point helps choose the correct half-plane.

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

Subtracting the two lines gives \(x=y\), then \(3x=8\), so \(x=y=\frac{8}{3}\). Find the intersection of slant boundaries separately.

Correct answer: A. ((5,0))

The boundary line is (2x-5y=10), and substituting ((5,0)) gives (10=10). For a boundary point, check equality.

Correct answer: A. When (x \le 2)

For both to hold, (x+2 \le 6-x), so (2x \le 4) and (x \le 2). For a region between two lines, the lower bound must not exceed the upper bound.

Correct answer: A. (24) square units

The line cuts the axes at ((8,0)) and ((0,6)), so area is (\frac{1}{2}\times 8\times 6=24). Intercepts quickly give the triangle area.

Correct answer: B. Unbounded region

In the first quadrant, (x+2y \ge 4) extends above the line without limit. If a region is not closed on all sides, it is unbounded.

Correct answer: C. ((5,2))

For ((5,2)), (x=5), which does not satisfy (x \le 4). If even one condition fails, the point lies outside.

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

The direct answer is option A: the intersection is (3,2). Solve the two boundary equations together. The second equation is x + y = 5. Multiply it by 2 to obtain 2x + 2y = 10. Subtract this from the first equation, 2x + 3y = 12: the result is y = 2. Put y = 2 into x + y = 5: x + 2 = 5, so x = 3. Therefore the intersection is (3,2). Check it in both equations: 2(3) + 3(2) = 12 and 3 + 2 = 5. Option A satisfies both. Option B, (1,4), gives 14 in the first equation. Option C, (4,1), gives 11 in the first equation. Option D, (2,3), gives 13 in the first equation. Thus the other points cannot be the intersection. Memory cue: an intersection point must satisfy every boundary equation simultaneously.

Correct answer: B. (y \ge \frac{x-6}{2})

From (x-2y \le 6), (-2y \le 6-x), so reversing the sign gives (y \ge \frac{x-6}{2}). Be careful when removing a negative coefficient.