Graphical solution of linear inequalities in two variables

Correct answer: A. (2x+y=6)

For a boundary line, replace the inequality sign by equality. In exams, draw the line first and then choose the region.

Correct answer: B. solid line

The boundary line for \(3x-2y\ge 12\) is \(3x-2y=12\). Since \(\ge\) includes equality, every point on the boundary is part of the solution region, so the line is drawn solid. A dashed line is used only for strict inequalities such as \(>\) or \(<\). Exam tip: use a solid line for \(\le\) and \(\ge\), and a dashed line for \(<\) and \(>\).

Correct answer: C. Draw \(2x+3y=6\) as a dashed line and shade the half-plane not containing the origin

Since the sign is \(>\), the boundary \(2x+3y=6\) is not included, so it is dashed. Testing the origin gives \(0>6\), which is false; hence shade the opposite half-plane. Exam tip: decide the boundary style before choosing the region.

Correct answer: B. no because (9\le 8) is false

Substituting ((1,4)) gives (1+8=9), which is greater than (8). In point testing, place both coordinates carefully.

Correct answer: C. \(6\)

The x-intercept is the point where the line meets the x-axis, so \(y=0\) there. Substituting \(y=0\) in \(x+3y=6\) gives \(x=6\). Therefore, the x-intercept is \(6\). The value \(3\) is related to the y-intercept, not the x-intercept. Exam tip: To find an x-intercept, always put \(y=0\).

Correct answer: A. \(8\)

To find the \(y\)-intercept, set \(x=0\). Substituting in \(4x+y=8\) gives \(4(0)+y=8\), so \(y=8\). Hence, the line meets the \(y\)-axis at \((0,8)\), and \(8\) is correct. \(4\) is only the coefficient of \(x\), not an intercept. Exam tip: always put \(x=0\) to find the \(y\)-intercept.

Correct answer: B. Above the line

To graph y ≥ 2x + 1, first draw the boundary line y = 2x + 1. The inequality says that y is greater than or equal to the value given by the line for the same x-coordinate. Therefore the solution consists of the half-plane above the line, and because equality is allowed, the boundary itself is included and is drawn as a solid line. A point test also confirms this: (0, 2) satisfies 2 ≥ 1 and lies above the line, whereas (0, 0) does not satisfy 0 ≥ 1. Thus option B is correct; option A represents the opposite inequality, option C omits valid points, and option D includes points that do not satisfy the condition.

Correct answer: A. first quadrant including axes

Both coordinates are non-negative, so the region is the first quadrant including axes. Always identify non-negativity conditions first.

Correct answer: B. left of \(x=4\), including the line

The inequality \(x\le 4\) means that the \(x\)-coordinate of every point is 4 or less. The boundary \(x=4\) is a vertical line, and it is included because the inequality contains \(\le\). Therefore, the required region is the half-plane to the left of the line, including the line. The region on the right represents \(x>4\). Exam tip: an inequality involving \(x\) has a vertical boundary line.

Correct answer: A. above (y=3) excluding the line

In (y>3), (y) is greater than (3) and the boundary is excluded. Shade above the dashed horizontal line.

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

Adding the equations gives (2x=8), so (x=4) and (y=2). Intersection points form corners of a feasible region.

Correct answer: A. (4, 2)

From the first line, y=10-2x. Substituting this in x+2y=8 gives x+2(10-2x)=8, so -3x=-12 and x=4. Then y=10-2(4)=2. Hence, the intersection point is (4, 2). For example, (2, 4) does not satisfy the first equation because 2(2)+4=8, not 10. Exam tip: Verify an intersection point by substituting it into both linear equations.

Correct answer: A. triangle

The axes and the line (x+2y=10) form a closed triangle. First-quadrant conditions can make the region bounded.

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

The intersection of (x=1) and (y=2) is ((1,2)), and it satisfies (x+y\le 7). To find vertices, intersect boundary lines pairwise.

Correct answer: B. Unbounded

The conditions x ≥ 0 and y ≥ 0 restrict the solution to the first quadrant. The inequality x + y ≥ 4 selects the side on or above the line x + y = 4. This region contains points such as (4, 0), (0, 4), (5, 0), and (0, 10), and it continues indefinitely as x or y increases. For example, every point of the form (t, 4) with t ≥ 0 satisfies all three inequalities, so there is no finite boundary enclosing the region. Hence the solution region is unbounded, making option B correct. It is not empty and is not a single point; it is also not bounded because it extends without limit.

Correct answer: C. (2, 2)

The governing concept is the intersection of solution sets of simultaneous linear inequalities. A point is in the feasible region only if it satisfies every condition: both coordinates must be non-negative and their sum must not exceed 3. Point A, (0,0), satisfies x ≥ 0, y ≥ 0, and 0 + 0 = 0 ≤ 3. Point B, (1,1), also satisfies the coordinate conditions and has sum 2 ≤ 3. Point C, (2,2), has non-negative coordinates, but its sum is 2 + 2 = 4, which is greater than 3; it violates the third inequality. Point D, (3,0), has sum 3 and is allowed because the boundary is included by ≤. Therefore option C is outside the region.

Correct answer: B. side opposite the origin

If the test point gives false, the solution region is not on that side. Hence choose the opposite half-plane.

Correct answer: B. false

For the origin test, put \(x=0\) and \(y=0\). The inequality \(x-2y\ge 6\) becomes \(0\ge 6\), which is false. Therefore, the origin does not lie in the solution half-plane, so the region on the side of the line \(x-2y=6\) opposite to the origin is selected. Option A uses the same inequality but incorrectly calls it true. Exam tip: after substituting a test point, always check whether the resulting inequality is true or false.

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

The x-intercept is found by putting y=0 in the equation: 2x=20, so x=10 and the point is (10,0). The y-intercept is found by putting x=0: 5y=20, so y=4 and the point is (0,4). Hence option A gives both intercepts. The other options interchange the values or fail to satisfy the original equation, so they cannot represent the line’s axis intercepts.

Correct answer: A. ((0,0)), ((8,0)), ((0,6))

The line intercepts are ((8,0)) and ((0,6)), and the axes meet at ((0,0)). These are the triangle vertices in the first quadrant.

Correct answer: A. left of (x=5) and below (y=3)

(x\le 5) gives the left side and (y\le 3) gives the lower side. The common region is the overlap of both conditions.

Correct answer: B. No

The governing distinction is between strict and non-strict inequalities. The solution x + 2y > 8 contains only points whose left-hand side is strictly greater than 8. Every point on the boundary line x + 2y = 8 gives equality, not a value greater than 8, so no boundary point satisfies the strict inequality. Consequently the boundary is excluded and would be drawn as a dashed line in a graph. Both listed points, (8, 0) and (0, 4), lie on that boundary because 8 + 2(0) = 8 and 0 + 2(4) = 8; neither is included. Therefore option B is correct, while option A and the two point-specific options incorrectly include equality cases.

Correct answer: A. yes

The symbol \(\ge\) means “greater than or equal to.” Hence, every point satisfying \(2x+3y=18\) also satisfies \(2x+3y\ge 18\). Therefore, the boundary line belongs to the solution region and is drawn as a solid line. With \(>\), the boundary would not be included and would be drawn dashed. Exam tip: use a solid boundary line for \(\ge\) or \(\le\).

Correct answer: C. (0, 3)

At the point of intersection, both equations must be satisfied. Substituting x=0 into x+4y=12 gives 4y=12, so y=3. Hence, the intersection is (0, 3). The point (3, 0) is the x-intercept, whereas x=0 gives a point on the y-axis. Exam tip: To find the y-intercept of a line, put x=0.

Correct answer: B. (4, 0)

The line y=0 represents the x-axis. Substituting y=0 in 5x+2y=20 gives 5x=20, so x=4. Hence, the intersection point is (4, 0). The point (0, 10) is the y-intercept of the line, not a point on the x-axis. Exam tip: To find an x-intercept, put y=0.