Graphical solution of linear inequalities in two variables

Correct answer: A. Bounded triangle in first quadrant

The feasible region is a bounded triangle in the first quadrant, so A is correct. The inequalities x≥0 and y≥0 restrict points to the first quadrant, including the axes. The inequality x+y≤8 means points on or below the line x+y=8 are allowed. This line meets the x-axis at (8,0) and the y-axis at (0,8), while the axes meet at (0,0). Therefore the region has vertices (0,0), (8,0), and (0,8), a closed bounded triangle. B is wrong because the second quadrant has x<0. C is wrong because x+y≤8 cuts off part of the first quadrant. D is wrong because (0,0), for example, satisfies all inequalities. Memory cue: nonnegative x and y mean first quadrant; a ≤ line often closes a bounded region.

Correct answer: A. (y<2x+3)

Below the line gives (y<), and exclusion of the line makes it strict. Check solid or dashed boundary in exams.

Correct answer: A. ((3,0)) and ((0,4))

Putting (y=0) gives (x=3), and putting (x=0) gives (y=4). The intercept method is quick for graphing.

Correct answer: A. First quadrant including axes

Both (x) and (y) are non-negative, so the region is the first quadrant. Do not miss non-negative restrictions in exams.

Correct answer: A. Empty region

It is impossible for (x+y) to be at most (2) and at least (5) at the same time. Identify contradictory conditions in exams.

Correct answer: A. Unbounded region

In the first quadrant, the region above (x+y=7) extends infinitely. Distinguish bounded and unbounded regions in exams.

Correct answer: A. Solid line

The boundary is a solid line, so A is correct. To draw the boundary of 5x−2y≤10, first replace the inequality sign by equality: 5x−2y=10. Because the symbol is ≤, points on the boundary are included in the solution, so the line must be drawn solid. For comparison, a strict sign < or > excludes the boundary and uses a dashed line. B is therefore wrong because a dashed line represents exclusion. C is wrong because the boundary of a two-variable linear inequality is a complete straight line, not merely separate points. D is wrong because the equation clearly supplies a boundary line. A test point such as (0,0) can then identify the shaded side, although it does not change the solid nature of the line. Memory cue: inclusive signs ≤ and ≥ mean solid; strict signs < and > mean dashed.

Correct answer: A. Rectangle

Both (x) and (y) lie in bounded intervals, so a rectangle is formed. Use vertical and horizontal lines for such inequalities.

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

The direct answer is option A: \((2,2)\). A vertex is a corner where boundary lines meet. The two relevant boundaries are \(2x+y=6\) and \(x+2y=6\). Subtracting the second equation from the first gives \(x-y=0\), so \(x=y\). Substituting into \(2x+y=6\) gives \(3x=6\), hence \(x=2\) and \(y=2\). The point satisfies \(x\ge0\), \(y\ge0\), and both upper-bound inequalities with equality, so it is feasible and is a vertex. Option B, \((4,1)\), gives \(2x+y=9>6\), so it is outside. Option C, \((1,4)\), gives \(x+2y=9>6\), so it is outside. Option D, \((3,3)\), violates both upper bounds. The axes also form other vertices, but among the choices only A is a vertex of the feasible region.

Correct answer: A. Left of the line

In (x<4), (x) is less than (4), so the left side is chosen. For vertical lines, think left-right.

Correct answer: A. On and above the line

For (y\ge -2), the boundary is included and the region above it is taken. For horizontal lines, check above-below.

Correct answer: A. The side containing origin is correct

The direct answer is option A: the side containing the origin is correct. The boundary of \(-x+y\le1\) is the line \(-x+y=1\). To decide which side to shade, test the origin \((0,0)\). Substitution gives \(-(0)+0=0\), and \(0\le1\) is true. Therefore the origin satisfies the inequality, so the half-plane containing the origin is the solution side. Option A is correct. Option B is wrong because it says the origin side is false, whereas the test calculation proves it true. Option C is wrong because the inequality includes an entire half-plane, not only its boundary; the boundary alone would represent an equality. Option D is wrong because many points satisfy the inequality, for example the origin itself. When the test point lies on the boundary, choose another point, but here the origin gives a clear result. Memory cue: substitute the test point and shade the side that makes the inequality true.

Correct answer: A. Any point not lying on the line

The direct answer is option A: choose a point that does not lie on the boundary line. To decide shading, substitute the coordinates of a test point into the inequality. A point on the boundary makes the two sides equal, so it only tells us that the point is on the line; it does not tell us which side satisfies the inequality. Therefore choose a convenient point away from the line, such as an intercept-free point that is easy to calculate with, and test it. Option A is correct because such a point lies clearly on one side and identifies the required region. Option B is wrong because the origin is not always suitable; here it lies on the boundary. Option C is wrong because
\((1,1)\) may also lie on the boundary and cannot decide the side. Option D is wrong because a test point is needed when the side is not already obvious. Memory cue: equality means boundary; use a point off the line for shading.

Correct answer: A. Empty region

For any (x), (y) cannot be both below (x+1) and above (x+4). Check parallel constraints in exams.

Correct answer: A. Strip between two parallel lines

Both lines have the same slope and the region lies between them. Use equal slopes to identify parallel lines.

Correct answer: A. On boundary and in solution

(2(5)+5(0)=10), so the point is on the boundary and included for (\ge). Check equality carefully.

Correct answer: A. Above the line

Because of (y\ge), the region above the line is taken and the line is included. The direction is clear in (y)-form.

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

The line (x+y=6) cuts the axes at ((6,0)) and ((0,6)). Write triangle vertices with axes in exams.

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

The direct answer is option A,
\((2,4)\). A feasible region contains only points that satisfy every inequality at the same time. Test option A step by step:
\(x=2\ge1\) is true,
\(y=4\ge2\) is true, and
\(x+y=2+4=6\le7\) is true. Thus it belongs to the feasible region. Option B,
\((0,4)\), satisfies the y-condition and the sum condition, but fails
\(x\ge1\) because
\(0<1\). Option C,
\((3,5)\), satisfies the first two conditions, but
\(3+5=8>7\), so it fails the third. Option D,
\((4,1)\), satisfies
\(x\ge1\), but fails
\(y\ge2\) because
\(1<2\). Therefore only A works. Exam cue: for a feasible point, check every inequality, not just most of them.

Correct answer: A. (x+2y\le 10)

At origin, (0\le 10) is true, and the solid line indicates (\le). Match both shading and line type.

Correct answer: A. \(y>2x\)

From \(2x-y<0\), we get \(-y<-2x\). Multiplying both sides by \(-1\) reverses the inequality sign, giving \(y>2x\). The form \(y<2x\) reverses the sign incorrectly, while \(y=2x\) represents only the boundary line and is not included in a strict inequality. Exam tip: always reverse an inequality sign when multiplying or dividing by a negative number.

Correct answer: A. Draw \(2x+y=4\) as a dashed line and shade the half-plane containing the origin

Since the sign is \(<\), the boundary \(2x+y=4\) is not included, so it must be dashed. Testing the origin gives \(2(0)+0=0<4\); therefore, shade the side containing the origin. Exam tip: decide dashed versus solid before choosing the region.

Correct answer: A. x=5 solid and y=1 dashed

The boundary-line rule depends on whether equality is included. For x≤5, the boundary is x=5, and the symbol ≤ includes equality, so x=5 must be drawn as a solid line. For y>1, the boundary is y=1, but the strict symbol > excludes equality, so y=1 must be drawn as a dashed line. Hence option A is correct. The shading for x≤5 is on the left side of the vertical line x=5, while the shading for y>1 is above the horizontal line y=1; their intersection is the solution region. Option B incorrectly treats the strict inequality as inclusive. Option C incorrectly excludes the boundary for x≤5. Option D reverses the two line types. Remember that the line style is determined by the inequality symbol, not by whether the boundary is vertical or horizontal.

Correct answer: A. \(ax+by\le c\)

A solid boundary means points satisfying \(ax+by=c\) are included. The shaded side has smaller values, so the inequality is \(ax+by\le c\). With \(<\), the boundary would be dashed. Exam tip: a solid line indicates equality.

Correct answer: A. (x\ge 0), (y\ge 0), (x+2y\le 4)

The small bounded region with axes is in the first quadrant and below the line. Use the origin test in exams.